# How do you differentiate f(x)=(x^2+4x)*secx using the product rule?

Dec 20, 2015

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

#### Explanation:

According to the product rule:

$f ' \left(x\right) = \sec x \frac{d}{\mathrm{dx}} \left[{x}^{2} + 4 x\right] + \left({x}^{2} + 4 x\right) \frac{d}{\mathrm{dx}} \left[\sec x\right]$

Find each derivative independently.

$\frac{d}{\mathrm{dx}} \left[{x}^{2} + 4 x\right] = 2 x + 4$

$\frac{d}{\mathrm{dx}} \left[\sec x\right] = \sec x \tan x$

Plug these values back in.

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

If you wish to factor:

$f ' \left(x\right) = \sec x \left(2 \left(x + 2\right) + x \left(x + 4\right) \tan x\right)$