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

Derivative of cos^3(x)

The derivative of

\cos^3(x)

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

.

Solution. Let

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

f(u) = u^3

, and

g(x) = \cos(x)

such that

F(x) = f(g(x))

. Then we will use the chain rule, i.e.:

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

We have seen here that

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

. Further, we know that

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

. So, we get

f'(u) = 3u^2

and therefore:

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

Wrapping everything, we get:

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

So, the derivative of

\cos^3(x)

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