Question

homework 3.6 the chain rule score: 80/100 answered: 8/10 question 9 textbook videos + let ( f(x)=6sin(cos x)) ( f(x)=)

Explanation:

Step1: Identify outer - inner functions

Let $u = \cos x$, then $y = 6\sin u$.

Step2: Differentiate outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=6\cos u$.

Step3: Differentiate inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=-\sin x$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$: $\frac{dy}{dx}=6\cos u\cdot(-\sin x)$. Replace $u$ with $\cos x$, we get $f^{\prime}(x)=- 6\sin x\cos(\cos x)$.

Answer:

$-6\sin x\cos(\cos x)$