Single Variable Calculus, Chapter 3, Review Exercises, Section Review Exercises, Problem 52

a.) Suppose that $f(x) = 4x - \tan x$, $\displaystyle \frac{-\pi}{2} < x < \frac{\pi}{2}$. Find $f'$ and $f''$

$\begin{equation} \begin{aligned} f'(x) &= 4 \frac{d}{dx} (x) - \frac{d}{dx} (\tan x)\\ \\ f'(x) &= 4 - \sec ^2 x\\ \\ f''(x) &= \frac{d}{dx} (4) - \frac{d}{dx} (\sec^2 x)\\ \\ f''(x) &= \frac{d}{dx} (4) - \frac{d}{dx} (\sec x)^2\\ \\ f''(x) &= 0 -2 (\sec x) \frac{d}{dx} (\sec x)\\ \\ f''(x) &= - 2 \sec x \sec x \tan x \\ \\ f''(x) &= -2 \sec^2 x \tan x \end{aligned} \end{equation}$


b.) Graph $f$, $f'$, and $f''$