What is the Derivative of sin^3(x)?

Derivative of sin^3(x)

The derivative of

\sin^3(x)

3\sin^2(x)\cos(x)

.

Solution. Let

F(x) = \sin^3(x)

f(u) = u^3

and

g(x) = \sin(x)

. We will use the chain rule:

\begin{align*} F'(x) = f'(g(x))g'(x). \end{align*}

We have seen here that

\frac{d}{dx} \sin(x) = \cos(x)

, and

\frac{d}{du} u^3 = 3u^2

. So we get:

\begin{align*} f'(u) = 3u^2 \quad \text{and} \quad g'(x) = \cos(x). \end{align*}

Together, we get:

\begin{align*} F'(x) &= f'(g(x))g'(x) \\ &= 3\sin^2(x)\cos(x). \end{align*}

So, the derivative of

\sin^3(x)

3\sin^2(x)\cos(x)