Question

I’m in AP Calculus AB and can’t figure this out. Can you help me solve this?

Answer 1

We need to derivate the next given function:

$f(x)=2x-x^2\tan x$

First, we have a subtraction, which means that we can derivate both expressions separately:

$\frac{d}{dx}f(x)=\frac{d}{dx}(2x)-\frac{d}{dx}(x^2\tan x)$

For 2x:

$\frac{d}{dx}(2x)=2$

for x²tanx, the expression represents a multiplication. Therefore, we use the next rule:

$\frac{d}{dx}(f\cdot g)=\frac{d}{dx}f\cdot g+f\cdot\frac{d}{dx}g$

Then:

$\frac{d}{dx}(x^2\tan x)=2x\tan x+x^2\sec ^2(x)$

The whole derivate would be:

$\frac{d}{dx}f(x)=2-(2x\tan x+x^2\sec ^2(x))$

Simplify the signs for the expression:

$\frac{d}{dx}f(x)=2-2x\tan x-x^2\sec ^2(x)$