# If f(x)=sec(x), how do I find f''(π/4)?

Mar 23, 2015

First find the expression for $f ' ' \left(x\right)$, then evaluate it at $\frac{\pi}{4}$.

$f \left(x\right) = \sec x$.

$f ' \left(x\right) = \sec x \tan x$.

Use the product rule to find $f ' ' \left(x\right)$.

$f ' ' \left(x\right) = \left(\sec x \tan x\right) \tan x + \sec x \left({\sec}^{2} x\right)$.

So, $f ' ' \left(x\right) = \sec x {\tan}^{2} x + {\sec}^{3} x$.

Evaluate:

$f ' ' \left(\frac{\pi}{4}\right) = \sec \left(\frac{\pi}{4}\right) {\tan}^{2} \left(\frac{\pi}{4}\right) + {\sec}^{3} \left(\frac{\pi}{4}\right)$

$= \left(\sqrt{2}\right) {\left(1\right)}^{2} + {\left(\sqrt{2}\right)}^{3} = \sqrt{2} + 2 \sqrt{2} = 3 \sqrt{2}$.