Question

How do you differentiate `sin(x^2)(cos(x^2))`?

Answer 1

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

Explanation:

This problem can also be solved by directly applying the differentiation rules:

`d/dx sin(x^2) cos(x^2)`

First, use the product rule:
`=sin(x^2) d/dx cos(x^2) + cos(x^2)d/dx sin(x^2)`

Then, each derivative can be solved using the chain rule:
`=-sin(x^2) sin(x^2) d/dx x^2 + cos(x^2)cos(x^2)d/dx x^2`

`=-2xsin^2(x^2) + 2xcos^2(x^2)`

At this point, we can simplify the expression by factoring out `2x` and applying the double angle identity `cos 2theta = cos^2 theta - sin^2 theta` :

`=2x(cos^2(x^2)-sin^2(x^2))`

`=2xcos(2x^2)`