Question

What is the derivative of `sinx(sinx+cosx)`?

Answer 1

The answer is `cos^2(x)-sin^2(x)+2cos(x)sin(x)=cos(2x)+sin(2x)`

Explanation:

First, use the Product Rule to say

`d/dx(sin(x)(sin(x)+cos(x)))=cos(x)(sin(x)+cos(x))+sin(x)(cos(x)-sin(x))`

Next, expand this out to write

`d/dx(sin(x)(sin(x)+cos(x)))=cos^2(x)-sin^2(x)+2cos(x)sin(x)`

Finally, use the double-angle formulas `cos^2(x)-sin^2(x)=cos(2x)` and `2cos(x)sin(x)=sin(2x)` to write

`d/dx(sin(x)(sin(x)+cos(x)))=cos(2x)+sin(2x)`