Question

How do you find the derivatives of the function `f(x) = sin(x^2)`?

Answer 1

Use the chain rule, which says, for the sine function:

`d/dx(sinu)=cosu* (du)/dx`

(Or, if you prefer function notation:

For `f(x) = sin(g(x))`, the derivative is:

`f'(x) = cos(g(x)) *g'(x)`

So for this `f(x) = sin(x^2)`, we get

`f'(x) = cos(x^2)*2x = 2xcos(x^2)`

The question asks about the derivatives (plural). I'm not sure what you mean.
Do you also need the second derivative?, or a formula for the `n^(th)` derivative?, or perhaps it was just an error typing the question.