for these 2 questions, how do you know they are differentiable? and how do you find the expressions?

Question

If f is a differentiable function, find an expression for the derivative of each of the following functions:

a) g(x) = f(x²) for this one I got 2x but I don't understand why it is supposed to be f(x²)(2x).

b) h(x) = 2xf(x) and for this one, the answer is 2xf(x) + 2f(x). why don't both of the x's go away in the 2f(x)? Also, how do you know if it is a plus sign or a multiplication sign in between those terms?

Thanks.

Answer 1

Part I

Chain Rule

g'(x)=2x f'(x²), i.e. 2x times the derivative of f(x²)

Part II

Product Rule

h'(x)=2xf'(x) + 2f(x)