Find the derivative of:

\begin{align}y(x) = \sin^3 x\end{align}

Step 1: Let \(\boldsymbol{u}\) be the nested function. Write the original function in terms of \(\boldsymbol{u}\).

In this case, the nested function is

\begin{align}u(x) = \sin x \end{align}

Therefore, our original function in terms of \(u\) is

\begin{align}y(u) = u^3\end{align}

\[\,\]

Step 2: Calculate the derivative of the original function with respect to \(\boldsymbol{u}\) and the derivative of \(\boldsymbol{u}\) with respect to \(\boldsymbol{x}\).

\begin{align}\frac{dy}{du} = 3u^2 && \frac{du}{dx} = \cos x\end{align}

\[\,\]

Step 3: Calculate the derivative of the original function with respect to \(\boldsymbol{x}\) using the chain rule.

\begin{align}\frac{dy}{dx} &= \frac{dy}{du} \frac{du}{dx}\\&=(3u^2)(\cos x)\\&=3u^2\cos x \end{align}

\[\,\]

Step 4: Reinsert what was defined in step 1 as \(\boldsymbol{u}\).

We now reinsert that \(u(x) = \sin x\) and obtain

\begin{align}\frac{dy}{dx} &=3u^2 \cos x \\&=3 \sin^2 x \cos x \end{align}