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

1 Answer
Jun 23, 2015

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)