Question

What is the derivative of `f(x)=cos(sin(5x))`?

Answer 1

`f'(x)=-5*sin(sin(5x))*cos(5x)`

Explanation:

To put it in simple words, when using the chain rule we differentiate what's outside, all that multiplied by what's inside (called function composition). Before solving your exercise, here's a little example:

`g(x)=sin(2x)`
`g'(x)=2cos(2x)`

So we can see the outside is the trigonometric function, and the inside is `2x`. The problem follows the same path, just a bit deeper, like inception. So:

`f(x)=cos(sin(5x))`
`f'(x)=d/dx(cos(sin(5x))`

The derivative of `cos(sin(5x))` is `-sin(sin(5x))`

Now, the differentiation of the inside, `sin(5x)`, is `cos(5x)` and the differentiation of what's inside is 5. Multiply all that and you get

`f'(x)=-sin(sin(5x))*cos(5x)*5`

If you want to rearrange:
`f'(x)=-5*sin(sin(5x))*cos(5x)`