Question

if $y = \sin^{3}x$, then $\frac{dy}{dx}=$
a $\cos^{3}x$
b $3\cos^{2}x$
c $3\sin^{2}x$
d $3\sin^{2}x\cos x$
e $3\sin^{2}x\cos x$

Explanation:

Step1: Identify the outer - inner functions

Let $u = \sin x$, then $y = u^{3}$.

Step2: Differentiate outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=3u^{2}$.

Step3: Differentiate inner function

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

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = \sin x$, $\frac{dy}{du}=3u^{2}$ and $\frac{du}{dx}=\cos x$ into the chain - rule formula, we get $\frac{dy}{dx}=3\sin^{2}x\cos x$.

Answer:

D. $3\sin^{2}x\cos x$