We are given the function:y=log\u20612(sec\u20612x)y = \\log_2 (\\sec^2 x)y=log2\u200b(sec2x)<\/p>\n\n\n\n
To find dydx\\frac{dy}{dx}dxdy\u200b, we’ll differentiate this with respect to xxx using the chain rule, properties of logarithms, and the derivatives of trigonometric functions.<\/p>\n\n\n\n
Step 1: Apply the change of base formula for logarithms<\/h3>\n\n\n\n
We can rewrite the logarithm in base 2 as:y=ln\u2061(sec\u20612x)ln\u20612y = \\frac{\\ln (\\sec^2 x)}{\\ln 2}y=ln2ln(sec2x)\u200b<\/p>\n\n\n\n
This simplifies the problem since we can now differentiate the natural logarithm of sec\u20612x\\sec^2 xsec2x.<\/p>\n\n\n\n
Step 2: Differentiate using the chain rule<\/h3>\n\n\n\n
We differentiate both sides of the equation with respect to xxx. First, apply the chain rule to differentiate ln\u2061(sec\u20612x)\\ln (\\sec^2 x)ln(sec2x):ddx(ln\u2061(sec\u20612x))=1sec\u20612x\u22c5ddx(sec\u20612x)\\frac{d}{dx} \\left( \\ln (\\sec^2 x) \\right) = \\frac{1}{\\sec^2 x} \\cdot \\frac{d}{dx} (\\sec^2 x)dxd\u200b(ln(sec2x))=sec2x1\u200b\u22c5dxd\u200b(sec2x)<\/p>\n\n\n\n
Now, differentiate sec\u20612x\\sec^2 xsec2x. Using the chain rule, we know:ddx(sec\u20612x)=2sec\u20612x\u22c5tan\u2061x\\frac{d}{dx} (\\sec^2 x) = 2 \\sec^2 x \\cdot \\tan xdxd\u200b(sec2x)=2sec2x\u22c5tanx<\/p>\n\n\n\n
Thus, we get:ddx(ln\u2061(sec\u20612x))=1sec\u20612x\u22c52sec\u20612x\u22c5tan\u2061x=2tan\u2061x\\frac{d}{dx} \\left( \\ln (\\sec^2 x) \\right) = \\frac{1}{\\sec^2 x} \\cdot 2 \\sec^2 x \\cdot \\tan x = 2 \\tan xdxd\u200b(ln(sec2x))=sec2x1\u200b\u22c52sec2x\u22c5tanx=2tanx<\/p>\n\n\n\n
Step 3: Combine and simplify<\/h3>\n\n\n\n
Now, putting this into our original equation:dydx=1ln\u20612\u22c52tan\u2061x\\frac{dy}{dx} = \\frac{1}{\\ln 2} \\cdot 2 \\tan xdxdy\u200b=ln21\u200b\u22c52tanx<\/p>\n\n\n\n
Thus, the derivative of yyy with respect to xxx is:dydx=2tan\u2061xln\u20612\\frac{dy}{dx} = \\frac{2 \\tan x}{\\ln 2}dxdy\u200b=ln22tanx\u200b<\/p>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n
dydx=2tan\u2061xln\u20612\\frac{dy}{dx} = \\frac{2 \\tan x}{\\ln 2}dxdy\u200b=ln22tanx\u200b<\/p>\n\n\n\n
This is the derivative of y=log\u20612(sec\u20612x)y = \\log_2 (\\sec^2 x)y=log2\u200b(sec2x) with respect to xxx. The key steps involved using the chain rule and properties of logarithms to simplify the problem.<\/p>\n\n\n\n If y = log2 sec2 x , find the derivative dy . dx The Correct Answer and Explanation is: We are given the function:y=log\u20612(sec\u20612x)y = \\log_2 (\\sec^2 x)y=log2\u200b(sec2x) To find dydx\\frac{dy}{dx}dxdy\u200b, we’ll differentiate this with respect to xxx using the chain rule, properties of logarithms, and the derivatives of trigonometric functions. Step 1: Apply the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","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":"","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-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":"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":""},"mobile":{"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":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-270710","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270710","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=270710"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270710\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=270710"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=270710"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=270710"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-1977.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"