Question

calculate the derivative of the following function. y = sin (9 cos x) $\frac{dy}{dx}=square$

Explanation:

Step1: Identify outer - inner functions

Let $u = 9\cos x$, then $y=\sin(u)$.

Step2: Differentiate outer function

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

Step3: Differentiate inner function

The derivative of $u = 9\cos x$ with respect to $x$ is $\frac{du}{dx}=-9\sin x$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\cos(u)$ and $\frac{du}{dx}=-9\sin x$ into the chain - rule formula. Replace $u$ with $9\cos x$, we get $\frac{dy}{dx}=\cos(9\cos x)\cdot(- 9\sin x)=-9\sin x\cos(9\cos x)$.

Answer:

$-9\sin x\cos(9\cos x)$