{"id":27084,"date":"2026-10-16T09:07:40","date_gmt":"2026-10-16T09:07:40","guid":{"rendered":"https:\/\/blog.oratrics.com\/?p=27084"},"modified":"2026-10-16T10:55:55","modified_gmt":"2026-10-16T10:55:55","slug":"derivative-of-square-root-of-3x","status":"publish","type":"post","link":"https:\/\/oratrics.com\/blogs\/uncategorized\/derivative-of-square-root-of-3x\/","title":{"rendered":"Derivative of Square Root of 3x: Formula, 3 Methods &amp; Solved Examples"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"27084\" class=\"elementor elementor-27084\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f936373 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"f936373\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-74d820b\" data-id=\"74d820b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-f25a211\" data-id=\"f25a211\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-41f33eb elementor-widget elementor-widget-heading\" data-id=\"41f33eb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Derivative of Square Root of 3x: Formula, 3 Methods &amp; Solved Examples<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c6f31c9 elementor-widget elementor-widget-text-editor\" data-id=\"c6f31c9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">You\u2019re working through a calculus problem, and there it is: \u221a(3x). A square root with a 3 tucked inside. Your pen hovers. Do you treat the 3 like a coefficient? Does it change the rule? Where does it go?<\/span><\/p><p><span style=\"font-weight: 400;\">Take a breath. This one is friendlier than it looks, and once you see how it works, you\u2019ll be able to handle a whole family of similar problems.<\/span><\/p><p><span style=\"font-weight: 400;\">In this guide, we\u2019ll go through the formula, three different ways to get there, a handful of solved examples, and the slip-ups that quietly cost students marks on exams.<\/span><\/p><p><span style=\"font-weight: 400;\">The derivative of the square root of 3x is:<\/span><\/p><p><b>d\/dx [\u221a(3x)] = 3 \/ (2\u221a(3x))<\/b><\/p><p><span style=\"font-weight: 400;\">You\u2019ll also see it written as \u221a3 \/ (2\u221ax), or as \u00bd\u00b7\u221a(3\/x). Don\u2019t panic if your textbook shows a different-looking version. They\u2019re all the same thing wearing different outfits, and we\u2019ll prove that in a minute. The formula works for every x greater than 0.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-dc99a34 elementor-align-center elementor-widget elementor-widget-button\" data-id=\"dc99a34\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/demo.oratrics.com\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Book a Free Demo Today<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-430fcac elementor-widget elementor-widget-image\" data-id=\"430fcac\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"647\" src=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-1024x647.webp\" class=\"attachment-large size-large wp-image-24739\" alt=\"Speech on value of time with a confident schoolboy, clock, and hourglass illustrating the importance of time management.\" srcset=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-1024x647.webp 1024w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-300x190.webp 300w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-768x485.webp 768w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-1536x970.webp 1536w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value.webp 1578w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-499df42\" data-id=\"499df42\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ae0ae31 elementor-widget elementor-widget-html\" data-id=\"ae0ae31\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t\t<link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\r\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Poppins:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\r\n\r\n<div class=\"toc-wrapper\">\r\n    \r\n    \r\n    <div class=\"custom-toc\" id=\"custom-toc\">\r\n        <div class=\"toc-title\">\r\n            \u2630 Table of Contents\r\n        <\/div>\r\n        <ul class=\"toc-list\" id=\"toc-list\"><\/ul>\r\n    <\/div>\r\n    \r\n<\/div>\r\n\r\n<style>\r\n\/* Parent Fixed Wrapper holding both components together *\/\r\n.toc-wrapper {\r\n    position: fixed;\r\n    top: 150px; \r\n    z-index: 1 !important;\r\n    right: 70px;\r\n    width: 100%;\r\n    max-width: 340px;\r\n    font-family: 'Poppins', sans-serif;\r\n}\r\n\r\n\/* Image Styling *\/\r\n.toc-sticky-image {\r\n    width: 100%;\r\n    height: auto;\r\n    border-radius: 22px; \r\n    margin-bottom: 15px; \r\n    box-shadow: 0 4px 12px rgba(0,0,0,0.05);\r\n    display: block;\r\n}\r\n\r\n\/* TOC Container *\/\r\n.custom-toc {\r\n    background: #fff;\r\n    border: 1px solid #dfe3ea;\r\n    border-radius: 22px;\r\n    padding: 0px 28px;\r\n    width: 100%;\r\n    max-height: 24rem; \r\n    overflow-y: auto;\r\n    box-shadow: 0 2px 8px rgba(0,0,0,0.03);\r\n}\r\n\r\n\/* Title *\/\r\n.custom-toc .toc-title {\r\n    font-size: 25px;\r\n    font-weight: 700;\r\n    color: #17233c;\r\n    position: sticky;\r\n    padding: 0.6rem 0rem;\r\n    background-color: white;\r\n    top: 0px;\r\n    margin-bottom: 12px;\r\n    display: flex;\r\n    align-items: center;\r\n    gap: 10px;\r\n    z-index: 2;\r\n}\r\n\r\n\/* List *\/\r\n.custom-toc .toc-list {\r\n    list-style: none;\r\n    margin: 0;\r\n    padding: 0;\r\n}\r\n\r\n\/* Links *\/\r\n.custom-toc .toc-list li {\r\n    margin-bottom: 22px;\r\n}\r\n\r\n.custom-toc .toc-list a {\r\n    text-decoration: none;\r\n    color: #6b7a99;\r\n    font-size: 14px;\r\n    font-weight: 600;\r\n    transition: 0.3s ease;\r\n    display: inline-block;\r\n}\r\n\r\n\/* Hover Color *\/\r\n.custom-toc .toc-list a:hover {\r\n    color: #8B0000;\r\n}\r\n\r\n\/* H2 Indent *\/\r\n.custom-toc .toc-list .toc-h2 {\r\n    padding-left: 18px;\r\n    font-size: 14px;\r\n}\r\n\r\n\/* Mobile Responsive Adjustments *\/\r\n@media(max-width:768px) {\r\n    .toc-wrapper {\r\n        position: relative;\r\n        top: 0;\r\n        z-index: 1;\r\n        right: 0;\r\n        max-width: 100%;\r\n        margin-bottom: 30px;\r\n    }\r\n\r\n    .custom-toc {\r\n        max-height: none; \r\n        overflow-y: visible;\r\n    }\r\n\r\n    .custom-toc .toc-title {\r\n        font-size: 25px;\r\n    }\r\n\r\n    .custom-toc .toc-list a {\r\n        font-size: 14px;\r\n    }\r\n}\r\n<\/style>\r\n\r\n<script>\r\ndocument.addEventListener(\"DOMContentLoaded\", function () {\r\n    const tocList = document.getElementById(\"toc-list\");\r\n    const h1 = document.querySelector(\"h1\");\r\n    const h2s = document.querySelectorAll(\"h2\");\r\n\r\n    function createId(text) {\r\n        return text.toLowerCase().replace(\/[^a-z0-9]+\/g, '-');\r\n    }\r\n\r\n    if (h1) {\r\n        if (!h1.id) {\r\n            h1.id = createId(h1.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.innerHTML = `<a href=\"#${h1.id}\">${h1.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    }\r\n\r\n    h2s.forEach(h2 => {\r\n        if (!h2.id) {\r\n            h2.id = createId(h2.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.classList.add(\"toc-h2\");\r\n        li.innerHTML = `<a href=\"#${h2.id}\">${h2.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    });\r\n});\r\n<\/script>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f27fe07 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"f27fe07\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-52a084c\" data-id=\"52a084c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a9d28a1\" data-id=\"a9d28a1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b4011b7 elementor-widget elementor-widget-heading\" data-id=\"b4011b7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Let\u2019s Look at What We\u2019re Dealing With<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1c12e98 elementor-widget elementor-widget-text-editor\" data-id=\"1c12e98\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Calculus rules like numbers more than they like roots. So the trick to differentiating almost any radical function is to turn the root into a power:<\/span><\/p><p><span style=\"font-weight: 400;\">\u221a(3x) = (3x)^(1\/2)<\/span><\/p><p><span style=\"font-weight: 400;\">That\u2019s it. That one move unlocks the power rule and the chain rule. A few reminders before we start, because we\u2019ll lean on them all the way through:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Power rule:<\/b><span style=\"font-weight: 400;\"> d\/dx [x\u207f] = n\u00b7x\u207f\u207b\u00b9<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Chain rule:<\/b><span style=\"font-weight: 400;\"> differentiate the outside, then multiply by the derivative of the inside<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Constant multiple rule:<\/b><span style=\"font-weight: 400;\"> a constant in front just tags along<\/span><\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a329014\" data-id=\"a329014\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0497ec5 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"0497ec5\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-97d8fce\" data-id=\"97d8fce\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3a0732e\" data-id=\"3a0732e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3659af7 elementor-widget elementor-widget-heading\" data-id=\"3659af7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Where does this derivative actually exist?\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a6e7fe4 elementor-widget elementor-widget-text-editor\" data-id=\"a6e7fe4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The function \u221a(3x) is defined for x \u2265 0. But its derivative only exists for x &gt; 0. At x = 0, the formula puts a zero in the denominator, and that\u2019s a no-go. If you sketch the graph, you\u2019ll see why: right at the origin the curve shoots straight up, so the tangent line is vertical, and a vertical line doesn\u2019t have a regular slope.<\/span><\/p><p><span style=\"font-weight: 400;\">It\u2019s a small detail, but exam questions love to ask about it.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-ad6e74a\" data-id=\"ad6e74a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b6f2417 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"b6f2417\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-347468c\" data-id=\"347468c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3fdc842\" data-id=\"3fdc842\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a0174a1 elementor-widget elementor-widget-heading\" data-id=\"a0174a1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Method 1: The Chain Rule<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-087e65d elementor-widget elementor-widget-text-editor\" data-id=\"087e65d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This is the method most teachers want to see, and it\u2019s the one worth mastering, because it works on far messier functions later.<\/span><\/p><p><span style=\"font-weight: 400;\">Start by writing y = (3x)^(1\/2). Think of it as two layers. The outer layer is \u201csomething to the power of \u00bd,\u201d and the inner layer is 3x.<\/span><\/p><p><span style=\"font-weight: 400;\">Differentiate the outer layer first, leaving the inside alone: (1\/2)\u00b7(3x)^(\u22121\/2).<\/span><\/p><p><span style=\"font-weight: 400;\">Then multiply by the derivative of the inside. The derivative of 3x is just 3.<\/span><\/p><p><span style=\"font-weight: 400;\">Put it together:<\/span><\/p><p><span style=\"font-weight: 400;\">dy\/dx = (1\/2)\u00b7(3x)^(\u22121\/2)\u00b73 = 3 \/ (2\u221a(3x))<\/span><\/p><p><span style=\"font-weight: 400;\">Notice that the 3 on top comes from that last step. This is the piece people forget, so keep an eye on it.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-576faaf\" data-id=\"576faaf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9d2d627 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"9d2d627\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-5856ed4\" data-id=\"5856ed4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-0d0713b\" data-id=\"0d0713b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0d47645 elementor-widget elementor-widget-heading\" data-id=\"0d47645\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Method 2: The Power Rule Shortcut\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1fe94e1 elementor-widget elementor-widget-text-editor\" data-id=\"1fe94e1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Here\u2019s a little trick I like because it avoids the chain rule altogether. Square roots split nicely over multiplication, so:<\/span><\/p><p><span style=\"font-weight: 400;\">\u221a(3x) = \u221a3 \u00b7 \u221ax<\/span><\/p><p><span style=\"font-weight: 400;\">And \u221a3 is just a number, about 1.732. A constant. So we can pull it out front and differentiate \u221ax the way we always do:<\/span><\/p><p><span style=\"font-weight: 400;\">d\/dx [\u221a3 \u00b7 x^(1\/2)] = \u221a3 \u00b7 (1\/2)\u00b7x^(\u22121\/2) = \u221a3 \/ (2\u221ax)<\/span><\/p><p><span style=\"font-weight: 400;\">Wait, is that really the same as 3\/(2\u221a(3x))? Check it. The denominator 2\u221a(3x) is 2\u00b7\u221a3\u00b7\u221ax, so we have 3 \/ (2\u00b7\u221a3\u00b7\u221ax). Since 3 = \u221a3\u00b7\u221a3, one \u221a3 cancels with the one on the bottom, leaving \u221a3 \/ (2\u221ax). Same answer.<\/span><\/p><p><span style=\"font-weight: 400;\">This shortcut is great when the inside is just a constant times x. It\u2019s less useful when the inside gets more complicated, like 3x + 1.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e47141f\" data-id=\"e47141f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0daa914 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"0daa914\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-5a6bd83\" data-id=\"5a6bd83\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-93c5352\" data-id=\"93c5352\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5ab0396 elementor-widget elementor-widget-heading\" data-id=\"5ab0396\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Method 3: First Principles (The \u201cWhy It Works\u201d Method)\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2dd7262 elementor-widget elementor-widget-text-editor\" data-id=\"2dd7262\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">If your teacher says \u201cprove it from the definition,\u201d this is what they mean. It\u2019s longer, but it\u2019s also the most satisfying, because nothing is taken on trust.<\/span><\/p><p><span style=\"font-weight: 400;\">The definition is:<\/span><\/p><p><span style=\"font-weight: 400;\">f\u2032(x) = lim (h\u21920) [f(x + h) \u2212 f(x)] \/ h<\/span><\/p><p><span style=\"font-weight: 400;\">For f(x) = \u221a(3x), that becomes:<\/span><\/p><p><span style=\"font-weight: 400;\">lim (h\u21920) [\u221a(3(x + h)) \u2212 \u221a(3x)] \/ h<\/span><\/p><p><span style=\"font-weight: 400;\">If you plug in h = 0 right away, you get 0\/0, which tells you nothing. So we use a classic move: multiply the top and bottom by the conjugate, \u221a(3(x + h)) + \u221a(3x).<\/span><\/p><p><span style=\"font-weight: 400;\">The top turns into a difference of squares:<\/span><\/p><p><span style=\"font-weight: 400;\">3(x + h) \u2212 3x = 3h<\/span><\/p><p><span style=\"font-weight: 400;\">So now we have:<\/span><\/p><p><span style=\"font-weight: 400;\">lim (h\u21920) 3h \/ { h \u00b7 [\u221a(3(x + h)) + \u221a(3x)] }<\/span><\/p><p><span style=\"font-weight: 400;\">The h on top and bottom cancels, which is exactly what we were hoping for:<\/span><\/p><p><span style=\"font-weight: 400;\">lim (h\u21920) 3 \/ [\u221a(3(x + h)) + \u221a(3x)]<\/span><\/p><p><span style=\"font-weight: 400;\">Now let h shrink to zero, and both square roots become \u221a(3x):<\/span><\/p><p><span style=\"font-weight: 400;\">3 \/ (2\u221a(3x))<\/span><\/p><p><span style=\"font-weight: 400;\">Three different roads, one destination. That\u2019s always a reassuring sign in math.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-388f227\" data-id=\"388f227\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-76fbba8 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"76fbba8\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-6f7cf12\" data-id=\"6f7cf12\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-b64fb00\" data-id=\"b64fb00\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d872166 elementor-widget elementor-widget-heading\" data-id=\"d872166\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Which Method Should You Use?<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1e42006 elementor-widget elementor-widget-text-editor\" data-id=\"1e42006\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Honestly, it depends on the moment.<\/span><\/p><p><span style=\"font-weight: 400;\">If you\u2019re doing homework or an exam and just need the answer, the chain rule is your best friend, because it scales up to anything. If the function is as simple as \u221a(3x), the power rule shortcut is the quickest. And if the question says \u201cfrom first principles\u201d or \u201cprove,\u201d you don\u2019t have a choice.<\/span><\/p><p><span style=\"font-weight: 400;\">If you only have time to really learn one, learn the chain rule.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1a7325b\" data-id=\"1a7325b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d00d198 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"d00d198\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-6e3e42b\" data-id=\"6e3e42b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-84cfcc6\" data-id=\"84cfcc6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b638c3e elementor-widget elementor-widget-heading\" data-id=\"b638c3e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">The Second Derivative<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8fd97e7 elementor-widget elementor-widget-text-editor\" data-id=\"8fd97e7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Sometimes you need to go one step further, for concavity or for motion problems. Start with the simplified first derivative:<\/span><\/p><p><span style=\"font-weight: 400;\">f\u2032(x) = (\u221a3 \/ 2)\u00b7x^(\u22121\/2)<\/span><\/p><p><span style=\"font-weight: 400;\">Differentiate once more:<\/span><\/p><p><span style=\"font-weight: 400;\">f\u2033(x) = (\u221a3 \/ 2)\u00b7(\u22121\/2)\u00b7x^(\u22123\/2) = \u2212\u221a3 \/ (4x^(3\/2))<\/span><\/p><p><span style=\"font-weight: 400;\">You may also see this as \u22129 \/ [4(3x)^(3\/2)]. Same thing.<\/span><\/p><p><span style=\"font-weight: 400;\">What does the negative sign tell us? The graph of \u221a(3x) is concave down for all x &gt; 0. It\u2019s still climbing, but the climb keeps getting gentler, like a hill that flattens out the farther you walk.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c329060\" data-id=\"c329060\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bbc2306 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"bbc2306\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-6d1e592\" data-id=\"6d1e592\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-5fd1bd2\" data-id=\"5fd1bd2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1ebf6d5 elementor-widget elementor-widget-heading\" data-id=\"1ebf6d5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Solved Examples<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6e7ebe6 elementor-widget elementor-widget-text-editor\" data-id=\"6e7ebe6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><b>Example 1: Finding a tangent line<\/b><\/p><p><span style=\"font-weight: 400;\">Find the slope of y = \u221a(3x) at x = 12, and write the tangent line.<\/span><\/p><p><span style=\"font-weight: 400;\">The derivative is 3\/(2\u221a(3x)). At x = 12, the inside is 36, and \u221a36 = 6, so the slope is 3\/12 = <\/span><b>1\/4<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">The point on the curve is (12, 6). The tangent line is y \u2212 6 = \u00bc(x \u2212 12), which simplifies to <\/span><b>y = x\/4 + 3<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Example 2: When it shows up in a product<\/b><\/p><p><span style=\"font-weight: 400;\">Differentiate f(x) = x\u00b2\u00b7\u221a(3x).<\/span><\/p><p><span style=\"font-weight: 400;\">You could use the product rule: 2x\u00b7\u221a(3x) + x\u00b2\u00b73\/(2\u221a(3x)). That\u2019s correct, but it\u2019s a bit clunky to simplify.<\/span><\/p><p><span style=\"font-weight: 400;\">Here\u2019s the cleaner route. Rewrite the function first: x\u00b2\u00b7\u221a3\u00b7x^(1\/2) = \u221a3\u00b7x^(5\/2). Then the power rule gives:<\/span><\/p><p><span style=\"font-weight: 400;\">f\u2032(x) = <\/span><b>(5\u221a3 \/ 2)\u00b7x^(3\/2)<\/b><\/p><p><span style=\"font-weight: 400;\">Both approaches give the same value. The lesson is worth remembering: rewriting before you differentiate can save you a lot of work.<\/span><\/p><p><b>Example 3: When it shows up in a quotient<\/b><\/p><p><span style=\"font-weight: 400;\">Differentiate g(x) = \u221a(3x) \/ x.<\/span><\/p><p><span style=\"font-weight: 400;\">With the quotient rule:<\/span><\/p><p><span style=\"font-weight: 400;\">g\u2032(x) = [ (3\/(2\u221a(3x)))\u00b7x \u2212 \u221a(3x) ] \/ x\u00b2<\/span><\/p><p><span style=\"font-weight: 400;\">Here\u2019s a nice simplification: 3x \/ (2\u221a(3x)) is the same as \u221a(3x)\/2. So the top becomes \u221a(3x)\/2 \u2212 \u221a(3x) = \u2212\u221a(3x)\/2, and:<\/span><\/p><p><span style=\"font-weight: 400;\">g\u2032(x) = \u2212\u221a(3x) \/ (2x\u00b2) = <\/span><b>\u2212\u221a3 \/ (2x^(3\/2))<\/b><\/p><p><span style=\"font-weight: 400;\">Want to double-check? Rewrite g(x) as \u221a3\u00b7x^(\u22121\/2) and use the power rule. You\u2019ll land in the same place.<\/span><\/p><p><b>Example 4: A real-world rate of change<\/b><\/p><p><span style=\"font-weight: 400;\">Water fills a container so that its height after t seconds is h(t) = \u221a(3t) centimeters. How fast is the water level rising at t = 27 seconds?<\/span><\/p><p><span style=\"font-weight: 400;\">The rate is h\u2032(t) = 3 \/ (2\u221a(3t)). At t = 27, we get \u221a81 = 9, so:<\/span><\/p><p><span style=\"font-weight: 400;\">h\u2032(27) = 3 \/ 18 = <\/span><b>1\/6 cm per second<\/b><span style=\"font-weight: 400;\"> (about 0.17)<\/span><\/p><p><span style=\"font-weight: 400;\">At that moment the water was 9 cm deep and still rising, just slowly. That matches what the negative second derivative told us earlier: the rate keeps dropping.<\/span><\/p><p><b>Example 5: A close cousin, \u221a(3x + 1)<\/b><\/p><p><span style=\"font-weight: 400;\">Differentiate y = \u221a(3x + 1) and find the slope at x = 1.<\/span><\/p><p><span style=\"font-weight: 400;\">The chain rule applies the same way. The outside gives 1\/(2\u221a(3x + 1)), and the inside, 3x + 1, has derivative 3. So:<\/span><\/p><p><span style=\"font-weight: 400;\">y\u2032 = 3 \/ (2\u221a(3x + 1))<\/span><\/p><p><span style=\"font-weight: 400;\">At x = 1, that\u2019s 3 \/ (2\u00b72) = <\/span><b>3\/4<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">Same pattern, slightly bigger inside. Once you spot it, a whole group of problems gets easier.<\/span><\/p><p><b>Example 6: Second derivative at a point<\/b><\/p><p><span style=\"font-weight: 400;\">Find f\u2033(3) for f(x) = \u221a(3x).<\/span><\/p><p><span style=\"font-weight: 400;\">Using f\u2033(x) = \u2212\u221a3 \/ (4x^(3\/2)):<\/span><\/p><p><span style=\"font-weight: 400;\">f\u2033(3) = \u2212\u221a3 \/ (4\u00b73\u221a3) = <\/span><b>\u22121\/12<\/b><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8551d66\" data-id=\"8551d66\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b98a0f6 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"b98a0f6\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-4bea6f0\" data-id=\"4bea6f0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-678991d\" data-id=\"678991d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a79749f elementor-widget elementor-widget-heading\" data-id=\"a79749f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Mistakes That Catch Almost Everyone\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2667c50 elementor-widget elementor-widget-text-editor\" data-id=\"2667c50\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Let\u2019s be honest about where things go wrong. These come up again and again.<\/span><\/p><p><b>Forgetting the 3 on top.<\/b><span style=\"font-weight: 400;\"> Writing 1\/(2\u221a(3x)) instead of 3\/(2\u221a(3x)) is the classic. The chain rule asks you to multiply by the derivative of the inside, and the inside here is 3x, whose derivative is 3.<\/span><\/p><p><b>Dropping the 3 completely.<\/b><span style=\"font-weight: 400;\"> Some students end up with 1\/(2\u221ax), as though the 3 never existed. A constant inside the root absolutely changes the answer.<\/span><\/p><p><b>Skipping steps.<\/b><span style=\"font-weight: 400;\"> It feels slow to write out every line, but jumping ahead is exactly how small sign and coefficient errors sneak in. Write it out and check whether your final form matches the answer key.<\/span><\/p><p><b>Forgetting the domain.<\/b><span style=\"font-weight: 400;\"> If a question asks about x = 0 or a negative x, the honest answer is that the derivative doesn\u2019t exist there.<\/span><\/p><p><b>Mixing up \u221a(3x) with 3\u221ax.<\/b><span style=\"font-weight: 400;\"> They look similar, but they aren\u2019t equal. \u221a(3x) = \u221a3\u00b7\u221ax, while 3\u221ax has a different constant out front. The derivative of 3\u221ax is 3\/(2\u221ax), which is not the same as \u221a3\/(2\u221ax).<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-997cdc6\" data-id=\"997cdc6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-890715a elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"890715a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-fea1cdf\" data-id=\"fea1cdf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8a12b52\" data-id=\"8a12b52\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-83565ea elementor-widget elementor-widget-heading\" data-id=\"83565ea\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Why bother? Where This Shows Up<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8d52ad6 elementor-widget elementor-widget-text-editor\" data-id=\"8d52ad6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Square root functions describe lots of things that grow quickly at first and then ease off: the height of liquid in certain containers, the speed of a falling object compared to the distance it\u2019s fallen, and some cost or output curves where each extra unit gives a bit less than the last. The derivative tells you how fast the quantity is changing right now, which is the whole point of calculus, and why physics, engineering, and economics courses keep coming back to it.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-d662da9\" data-id=\"d662da9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e8b28f3 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"e8b28f3\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-737a6c2\" data-id=\"737a6c2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-b4d776f\" data-id=\"b4d776f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4fcc9e8 elementor-widget elementor-widget-heading\" data-id=\"4fcc9e8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Quick Revision Sheet<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1f83ffb elementor-widget elementor-widget-text-editor\" data-id=\"1f83ffb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">If you\u2019re short on time, here\u2019s the essence:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u221a(3x) = (3x)^(1\/2)<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">First derivative: 3 \/ (2\u221a(3x)), also \u221a3 \/ (2\u221ax)<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Second derivative: \u2212\u221a3 \/ (4x^(3\/2))<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The derivative exists only for x &gt; 0<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Tools to know: chain rule, power rule, first principles<\/span><\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-4dc00ae\" data-id=\"4dc00ae\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-940856b elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"940856b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-2fd75bf\" data-id=\"2fd75bf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3dad9e8\" data-id=\"3dad9e8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1829d87 elementor-widget elementor-widget-heading\" data-id=\"1829d87\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Conclusion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9a54b2c elementor-widget elementor-widget-text-editor\" data-id=\"9a54b2c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The derivative of the square root of 3x really comes down to two habits: rewrite the root as a power, and never forget the derivative of what\u2019s inside. Try a few variations on your own, like \u221a(3x + 5), \u221a(3x\u00b2), or a product with x\u00b3, and you\u2019ll find the pattern sticks faster than you\u2019d expect. And if you ever doubt an answer, run it through a second method. When two routes agree, you can trust the result.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-733ff78\" data-id=\"733ff78\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d43f874 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"d43f874\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-77aa3c9\" data-id=\"77aa3c9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-0e4ebe1\" data-id=\"0e4ebe1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4ef5223 elementor-widget elementor-widget-heading\" data-id=\"4ef5223\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Frequently Asked Questions\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-34132af elementor-widget elementor-widget-accordion\" data-id=\"34132af\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"accordion.default\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-5461\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-5461\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">What is the derivative of the square root of 3x?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-5461\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-5461\"><p><span style=\"font-weight: 400;\">It\u2019s 3 \/ (2\u221a(3x)), which can also be written as \u221a3 \/ (2\u221ax).<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-5462\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-5462\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">How do I find the derivative of root 3x with the power rule?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-5462\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-5462\"><p><span style=\"font-weight: 400;\">Rewrite \u221a(3x) as \u221a3\u00b7x^(1\/2), pull out \u221a3, and differentiate x^(1\/2) to get \u221a3 \/ (2\u221ax).<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-5463\" class=\"elementor-tab-title\" data-tab=\"3\" role=\"button\" aria-controls=\"elementor-tab-content-5463\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Does the derivative of \u221a(3x) exist at x = 0?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-5463\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"3\" role=\"region\" aria-labelledby=\"elementor-tab-title-5463\"><p><span style=\"font-weight: 400;\">No. The function itself is defined there, but the slope becomes infinite, so the derivative doesn\u2019t exist.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-5464\" class=\"elementor-tab-title\" data-tab=\"4\" role=\"button\" aria-controls=\"elementor-tab-content-5464\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Why is there a 3 in the numerator?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-5464\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"4\" role=\"region\" aria-labelledby=\"elementor-tab-title-5464\"><p><span style=\"font-weight: 400;\">It comes from the chain rule. The derivative of the inner function 3x is 3, and it multiplies the outer derivative.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-5465\" class=\"elementor-tab-title\" data-tab=\"5\" role=\"button\" aria-controls=\"elementor-tab-content-5465\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Are \u221a(3x) and \u221a(3\/x) differentiated the same way?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-5465\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"5\" role=\"region\" aria-labelledby=\"elementor-tab-title-5465\"><p><span style=\"font-weight: 400;\">Not quite. \u221a(3\/x) = \u221a3\u00b7x^(\u22121\/2), and its derivative is \u2212\u221a3 \/ (2x^(3\/2)). Notice the negative sign and the different power.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"What is the derivative of the square root of 3x?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">It\\u2019s 3 \\\/ (2\\u221a(3x)), which can also be written as \\u221a3 \\\/ (2\\u221ax).<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"How do I find the derivative of root 3x with the power rule?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">Rewrite \\u221a(3x) as \\u221a3\\u00b7x^(1\\\/2), pull out \\u221a3, and differentiate x^(1\\\/2) to get \\u221a3 \\\/ (2\\u221ax).<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"Does the derivative of \\u221a(3x) exist at x = 0?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">No. The function itself is defined there, but the slope becomes infinite, so the derivative doesn\\u2019t exist.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"Why is there a 3 in the numerator?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">It comes from the chain rule. The derivative of the inner function 3x is 3, and it multiplies the outer derivative.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"Are \\u221a(3x) and \\u221a(3\\\/x) differentiated the same way?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">Not quite. \\u221a(3\\\/x) = \\u221a3\\u00b7x^(\\u22121\\\/2), and its derivative is \\u2212\\u221a3 \\\/ (2x^(3\\\/2)). Notice the negative sign and the different power.<\\\/span><\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-de28a01\" data-id=\"de28a01\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Derivative of Square Root of 3x: Formula, 3 Methods &amp; Solved Examples You\u2019re working through a calculus problem, and there [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[1],"tags":[],"class_list":["post-27084","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Derivative of Square Root of 3x: Formula, 3 Methods &amp; 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