Second derivative of ln(sec x)

Let $f(x) = \ln(\sec x)$ . Find $f''(x)$ .

Hint

First find $f'(x)$ using the chain rule, recalling that the derivative of $\sec x$ is $\sec x \tan x$ . Simplify $f'(x)$ before differentiating again.

Step-by-step solution

By the chain rule, $f'(x) = \dfrac{1}{\sec x}\cdot \sec x \tan x = \tan x$ . This simplification makes the next differentiation much easier.

Now differentiate $f'(x) = \tan x$ to get $f''(x) = \sec^2 x$ , using the known derivative of tangent.

Second derivative of ln(sec x)

Related problems