To find the derivative of the function 4\u2212sec\u2061(x)tan\u2061(x)\\frac{4 – \\sec(x)}{\\tan(x)}tan(x)4\u2212sec(x)\u200b, we’ll use the quotient rule, which is applied to functions of the form u(x)v(x)\\frac{u(x)}{v(x)}v(x)u(x)\u200b.<\/p>\n\n\n\n
The quotient rule states: ddx(u(x)v(x))=v(x)\u22c5u\u2032(x)\u2212u(x)\u22c5v\u2032(x)(v(x))2\\frac{d}{dx} \\left( \\frac{u(x)}{v(x)} \\right) = \\frac{v(x) \\cdot u'(x) – u(x) \\cdot v'(x)}{(v(x))^2}dxd\u200b(v(x)u(x)\u200b)=(v(x))2v(x)\u22c5u\u2032(x)\u2212u(x)\u22c5v\u2032(x)\u200b<\/p>\n\n\n\n
Where u(x)=4\u2212sec\u2061(x)u(x) = 4 – \\sec(x)u(x)=4\u2212sec(x) and v(x)=tan\u2061(x)v(x) = \\tan(x)v(x)=tan(x).<\/p>\n\n\n\n
Step 1: Differentiate u(x)=4\u2212sec\u2061(x)u(x) = 4 – \\sec(x)u(x)=4\u2212sec(x)<\/h3>\n\n\n\n
The derivative of a constant is zero, so the derivative of 4 is 0. To differentiate \u2212sec\u2061(x)-\\sec(x)\u2212sec(x), we use the chain rule: ddxsec\u2061(x)=sec\u2061(x)tan\u2061(x)\\frac{d}{dx} \\sec(x) = \\sec(x) \\tan(x)dxd\u200bsec(x)=sec(x)tan(x)<\/p>\n\n\n\n
Thus, the derivative of u(x)u(x)u(x) is: u\u2032(x)=\u2212sec\u2061(x)tan\u2061(x)u'(x) = -\\sec(x) \\tan(x)u\u2032(x)=\u2212sec(x)tan(x)<\/p>\n\n\n\n
Step 2: Differentiate v(x)=tan\u2061(x)v(x) = \\tan(x)v(x)=tan(x)<\/h3>\n\n\n\n
The derivative of tan\u2061(x)\\tan(x)tan(x) is sec\u20612(x)\\sec^2(x)sec2(x), so: v\u2032(x)=sec\u20612(x)v'(x) = \\sec^2(x)v\u2032(x)=sec2(x)<\/p>\n\n\n\n
Step 3: Apply the quotient rule<\/h3>\n\n\n\n
Now we apply the quotient rule: ddx(4\u2212sec\u2061(x)tan\u2061(x))=tan\u2061(x)\u22c5(\u2212sec\u2061(x)tan\u2061(x))\u2212(4\u2212sec\u2061(x))\u22c5sec\u20612(x)tan\u20612(x)\\frac{d}{dx} \\left( \\frac{4 – \\sec(x)}{\\tan(x)} \\right) = \\frac{\\tan(x) \\cdot (-\\sec(x) \\tan(x)) – (4 – \\sec(x)) \\cdot \\sec^2(x)}{\\tan^2(x)}dxd\u200b(tan(x)4\u2212sec(x)\u200b)=tan2(x)tan(x)\u22c5(\u2212sec(x)tan(x))\u2212(4\u2212sec(x))\u22c5sec2(x)\u200b<\/p>\n\n\n\n
Step 4: Simplify the expression<\/h3>\n\n\n\n
We can simplify each term in the numerator: =\u2212sec\u2061(x)tan\u20612(x)\u2212(4\u2212sec\u2061(x))sec\u20612(x)tan\u20612(x)= \\frac{-\\sec(x) \\tan^2(x) – (4 – \\sec(x)) \\sec^2(x)}{\\tan^2(x)}=tan2(x)\u2212sec(x)tan2(x)\u2212(4\u2212sec(x))sec2(x)\u200b<\/p>\n\n\n\n
Distribute sec\u20612(x)\\sec^2(x)sec2(x) in the second term: =\u2212sec\u2061(x)tan\u20612(x)\u22124sec\u20612(x)+sec\u20613(x)tan\u20612(x)= \\frac{-\\sec(x) \\tan^2(x) – 4\\sec^2(x) + \\sec^3(x)}{\\tan^2(x)}=tan2(x)\u2212sec(x)tan2(x)\u22124sec2(x)+sec3(x)\u200b<\/p>\n\n\n\n
Final answer:<\/h3>\n\n\n\n
The derivative of 4\u2212sec\u2061(x)tan\u2061(x)\\frac{4 – \\sec(x)}{\\tan(x)}tan(x)4\u2212sec(x)\u200b is: \u2212sec\u2061(x)tan\u20612(x)\u22124sec\u20612(x)+sec\u20613(x)tan\u20612(x)\\frac{-\\sec(x) \\tan^2(x) – 4 \\sec^2(x) + \\sec^3(x)}{\\tan^2(x)}tan2(x)\u2212sec(x)tan2(x)\u22124sec2(x)+sec3(x)\u200b<\/p>\n\n\n\n
This expression involves standard trigonometric derivatives and simplifications. It gives you the rate of change of the function 4\u2212sec\u2061(x)tan\u2061(x)\\frac{4 – \\sec(x)}{\\tan(x)}tan(x)4\u2212sec(x)\u200b with respect to xxx.<\/p>\n\n\n\n Derivative of (4-sec(x))\/(tan(x)) The Correct Answer and Explanation is: To find the derivative of the function 4\u2212sec\u2061(x)tan\u2061(x)\\frac{4 – \\sec(x)}{\\tan(x)}tan(x)4\u2212sec(x)\u200b, we’ll use the quotient rule, which is applied to functions of the form u(x)v(x)\\frac{u(x)}{v(x)}v(x)u(x)\u200b. The quotient rule states: ddx(u(x)v(x))=v(x)\u22c5u\u2032(x)\u2212u(x)\u22c5v\u2032(x)(v(x))2\\frac{d}{dx} \\left( \\frac{u(x)}{v(x)} \\right) = \\frac{v(x) \\cdot u'(x) – u(x) \\cdot v'(x)}{(v(x))^2}dxd\u200b(v(x)u(x)\u200b)=(v(x))2v(x)\u22c5u\u2032(x)\u2212u(x)\u22c5v\u2032(x)\u200b Where u(x)=4\u2212sec\u2061(x)u(x) = 4 – \\sec(x)u(x)=4\u2212sec(x) […]<\/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-257758","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257758","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=257758"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257758\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=257758"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=257758"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=257758"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-829.jpeg\")
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