Question

question 15
textbook
let
$f(x)=-3\cos ^{5}x$
$f(x)=$
question help: video

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(y = f(g(x))\), then \(y^\prime=f^\prime(g(x))\cdot g^\prime(x)\). Let \(u = \cos x\), so \(f(x)=- 3u^{5}\). The derivative of \(f(u)\) with respect to \(u\) is \(f^\prime(u)=-3\times5u^{4}=-15u^{4}\), and the derivative of \(u = \cos x\) with respect to \(x\) is \(u^\prime=-\sin x\).

Step2: Substitute back

By the chain rule \(f^\prime(x)=f^\prime(u)\cdot u^\prime\). Substituting \(u = \cos x\), \(f^\prime(u)=-15u^{4}\) and \(u^\prime =-\sin x\) into the formula, we get \(f^\prime(x)=-15\cos^{4}x\times(-\sin x)=15\cos^{4}x\sin x\).

Answer:

$15\cos^{4}x\sin x$