What is the Derivative of sec^2(x)?

derivative of sec^2(x)

The derivative of

\sec^2(x)

2\tan(x)\sec^2(x)

.

Solution. Let

F(x) = \sec^2(x)

f(u) = u^2

and

g(x) = \sec(x)

. Then we will apply the chain rule:

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

We have seen here that

\frac{d}{dx} \sec(x) = \tan(x)\sec(x)

. So we get:

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

Combining everything, we get:

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

Therefore, the derivative of

\sec^2(x)

2\tan(x)\sec^2(x)