Derivative of sec^3(x)

The Mathematician — Tue, 02 May 2023 13:00:46 +0000

What is the derivative of $\sec^3(x)$ ?

The derivative of

\sec^3(x)

3\tan(x)\sec^3(x)

Let

F(x) = f(g(x)) = \sec^3(x)

, where

f(u) = u^3

and

g(x) = \sec(x)

. Then to determine the derivative of

\sec^3(x)

, we need to apply the chain rule:

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

It is easy to see that

f'(u) = 3u^2

and we have seen here that

g'(x) = \tan(x)\sec(x)

. So we get:

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

Substituting everything, we get:

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

So, the derivative of

\sec^3(x)

3\tan(x)\sec^3(x)