### Problem 4e

4e. Find the first and second derivative.

We first rewrite

and use the quotient rule.

This is often written

= sec^{2} x
For the second derivative, let us rewrite the first derivative as

f'(x) = (cos x)^{-2}
and use the power rule. Move the power in front of the
parentheses, and raise the quantity inside the parentheses to one
smaller power, and multiply everything by the derivative of the
inside function.

f"(x) = -2(cos x)^{-3}(-sin x)
This can be cleaned up to

which can also be written as

= 2sec^{2} x tan x