if (y = sin^{3}x), then (\frac{dy}{dx}=)

Explanation:

Step1: Apply chain - rule

Let $u = \sin x$, then $y = u^{3}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

Step2: Differentiate $y$ with respect to $u$

If $y = u^{3}$, then $\frac{dy}{du}=3u^{2}$.

Step3: Differentiate $u$ with respect to $x$

If $u=\sin x$, then $\frac{du}{dx}=\cos x$.

Step4: Substitute $u$ back and find $\frac{dy}{dx}$

Substitute $u = \sin x$ into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$. We get $\frac{dy}{dx}=3\sin^{2}x\cdot\cos x$.

Answer:

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