![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/image-351.png\")
<\/figure>\n\n\n\nThe Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n Correct Answer:<\/strong> Explanation:<\/strong><\/p>\n\n\n\n To find the derivative of the function f(x) = (2x\u00b3 + e^x) \/ sin\u00b2(x), we must use the Quotient Rule<\/strong>. The function is a quotient of two simpler functions. Let’s define them as:<\/p>\n\n\n\n The Quotient Rule states that for a function f(x) = g(x) \/ h(x), its derivative is: Step 1: Find the derivative of the numerator, g'(x).<\/strong> Step 2: Find the derivative of the denominator, h'(x).<\/strong> Step 3: Apply the Quotient Rule.<\/strong> Step 4: Simplify the expression.<\/strong> We can see a common factor of sin(x) in both terms of the numerator. Let’s factor it out: Now, we can cancel one sin(x) from the numerator and the denominator: Finally, distribute the 2 in the second term of the numerator: This result matches the second option provided.<\/p>\n\n\n\n QUESTION 2 Find the derivative of the function . The Correct Answer and Explanation is: Correct Answer:The correct option is the second one:(6x\u00b2 + e^x) sinx – (4x\u00b3 + 2e^x) cosx \/ sin\u00b3x Explanation: To find the derivative of the function f(x) = (2x\u00b3 + e^x) \/ sin\u00b2(x), we must use the Quotient Rule. The function […]<\/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-231282","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231282","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=231282"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231282\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=231282"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=231282"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=231282"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
The correct option is the second one:
(6x\u00b2 + e^x) sinx – (4x\u00b3 + 2e^x) cosx \/ sin\u00b3x<\/p>\n\n\n\n\n
f'(x) = [g'(x)h(x) – g(x)h'(x)] \/ [h(x)]\u00b2<\/p>\n\n\n\n
Using the power rule for 2x\u00b3 and the rule for the exponential function e^x:
g'(x) = d\/dx (2x\u00b3 + e^x) = 6x\u00b2 + e^x<\/p>\n\n\n\n
The function h(x) = sin\u00b2(x) can be written as (sin(x))\u00b2. We must use the Chain Rule<\/strong>.
Let u = sin(x), so h(x) = u\u00b2.
The derivative is h'(x) = (d\/du)(u\u00b2) * (d\/dx)(sin(x)).
h'(x) = (2u) * (cos(x))
Substituting u = sin(x) back into the equation:
h'(x) = 2sin(x)cos(x)<\/p>\n\n\n\n
Now, we substitute g(x), h(x), g'(x), and h'(x) into the Quotient Rule formula:
f'(x) = [ (6x\u00b2 + e^x) * sin\u00b2(x) – (2x\u00b3 + e^x) * (2sin(x)cos(x)) ] \/ [sin\u00b2(x)]\u00b2<\/p>\n\n\n\n
The denominator becomes (sin\u00b2(x))\u00b2 = sin\u2074(x).
f'(x) = [ (6x\u00b2 + e^x)sin\u00b2(x) – 2(2x\u00b3 + e^x)sin(x)cos(x) ] \/ sin\u2074(x)<\/p>\n\n\n\n
f'(x) = [ sin(x) * { (6x\u00b2 + e^x)sin(x) – 2(2x\u00b3 + e^x)cos(x) } ] \/ sin\u2074(x)<\/p>\n\n\n\n
f'(x) = [ (6x\u00b2 + e^x)sin(x) – 2(2x\u00b3 + e^x)cos(x) ] \/ sin\u00b3(x)<\/p>\n\n\n\n
f'(x) = [ (6x\u00b2 + e^x)sin(x) – (4x\u00b3 + 2e^x)cos(x) ] \/ sin\u00b3(x)<\/p>\n\n\n\n![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner4-875.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"