{"id":270145,"date":"2025-07-25T05:07:56","date_gmt":"2025-07-25T05:07:56","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=270145"},"modified":"2025-07-25T05:07:59","modified_gmt":"2025-07-25T05:07:59","slug":"find-the-derivative-of-y-2x-cos-x-2-sin-x-please-show-all-work","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/25\/find-the-derivative-of-y-2x-cos-x-2-sin-x-please-show-all-work\/","title":{"rendered":"Find the derivative of y = 2x cos x – 2 sin x Please show all work."},"content":{"rendered":"\n

Find the derivative of y = 2x cos x – 2 sin x Please show all work.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To find the derivative of the function y=2xcos\u2061x\u22122sin\u2061xy = 2x \\cos x – 2 \\sin xy=2xcosx\u22122sinx, we will apply the rules of differentiation, specifically the product rule and the standard derivatives of trigonometric functions.<\/p>\n\n\n\n

Step 1: Identify the components<\/h3>\n\n\n\n

The function is y=2xcos\u2061x\u22122sin\u2061xy = 2x \\cos x – 2 \\sin xy=2xcosx\u22122sinx. This is the sum of two terms, so we will differentiate each term separately.<\/p>\n\n\n\n

Step 2: Differentiate the first term 2xcos\u2061x2x \\cos x2xcosx<\/h3>\n\n\n\n

This term is a product of two functions, 2x2x2x and cos\u2061x\\cos xcosx. To differentiate a product, we use the product rule<\/strong>: ddx(u\u22c5v)=u\u2032v+uv\u2032\\frac{d}{dx}(u \\cdot v) = u’v + uv’dxd\u200b(u\u22c5v)=u\u2032v+uv\u2032<\/p>\n\n\n\n

Here, let u=2xu = 2xu=2x and v=cos\u2061xv = \\cos xv=cosx. Now, differentiate each part:<\/p>\n\n\n\n

    \n
  • The derivative of u=2xu = 2xu=2x is u\u2032=2u’ = 2u\u2032=2.<\/li>\n\n\n\n
  • The derivative of v=cos\u2061xv = \\cos xv=cosx is v\u2032=\u2212sin\u2061xv’ = -\\sin xv\u2032=\u2212sinx (using the standard derivative of cos\u2061x\\cos xcosx).<\/li>\n<\/ul>\n\n\n\n

    Applying the product rule: ddx(2xcos\u2061x)=(2)\u22c5(cos\u2061x)+(2x)\u22c5(\u2212sin\u2061x)=2cos\u2061x\u22122xsin\u2061x\\frac{d}{dx}(2x \\cos x) = (2) \\cdot (\\cos x) + (2x) \\cdot (-\\sin x) = 2 \\cos x – 2x \\sin xdxd\u200b(2xcosx)=(2)\u22c5(cosx)+(2x)\u22c5(\u2212sinx)=2cosx\u22122xsinx<\/p>\n\n\n\n

    Step 3: Differentiate the second term \u22122sin\u2061x-2 \\sin x\u22122sinx<\/h3>\n\n\n\n

    This is a simple trigonometric function. The derivative of sin\u2061x\\sin xsinx is cos\u2061x\\cos xcosx, so: ddx(\u22122sin\u2061x)=\u22122cos\u2061x\\frac{d}{dx}(-2 \\sin x) = -2 \\cos xdxd\u200b(\u22122sinx)=\u22122cosx<\/p>\n\n\n\n

    Step 4: Combine the derivatives<\/h3>\n\n\n\n

    Now, we combine the derivatives of the two terms: dydx=(2cos\u2061x\u22122xsin\u2061x)\u22122cos\u2061x\\frac{dy}{dx} = (2 \\cos x – 2x \\sin x) – 2 \\cos xdxdy\u200b=(2cosx\u22122xsinx)\u22122cosx<\/p>\n\n\n\n

    Simplifying the expression: dydx=2cos\u2061x\u22122xsin\u2061x\u22122cos\u2061x\\frac{dy}{dx} = 2 \\cos x – 2x \\sin x – 2 \\cos xdxdy\u200b=2cosx\u22122xsinx\u22122cosx<\/p>\n\n\n\n

    The 2cos\u2061x2 \\cos x2cosx terms cancel out: dydx=\u22122xsin\u2061x\\frac{dy}{dx} = -2x \\sin xdxdy\u200b=\u22122xsinx<\/p>\n\n\n\n

    Final Answer:<\/h3>\n\n\n\n

    dydx=\u22122xsin\u2061x\\frac{dy}{dx} = -2x \\sin xdxdy\u200b=\u22122xsinx<\/p>\n\n\n\n

    Explanation:<\/h3>\n\n\n\n

    The derivative of y=2xcos\u2061x\u22122sin\u2061xy = 2x \\cos x – 2 \\sin xy=2xcosx\u22122sinx is found by using the product rule for the first term and the standard derivative for the second term. After simplifying, we get the result dydx=\u22122xsin\u2061x\\frac{dy}{dx} = -2x \\sin xdxdy\u200b=\u22122xsinx. This shows how we break down a more complex function into manageable pieces and apply differentiation rules step by step.<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Find the derivative of y = 2x cos x – 2 sin x Please show all work. The Correct Answer and Explanation is: To find the derivative of the function y=2xcos\u2061x\u22122sin\u2061xy = 2x \\cos x – 2 \\sin xy=2xcosx\u22122sinx, we will apply the rules of differentiation, specifically the product rule and the standard derivatives of […]<\/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-270145","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270145","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=270145"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270145\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=270145"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=270145"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=270145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}