We know that f(x)=
x^2
∫g(t)dt
0
What is f'(x) and which conditions are necessary for g(t)?
By the "Fundamental theorem of Calculus", the derivative of $\displaystyle \int_a^u g(t) dt$, with respect to u, is g(u). Here, $\displaystyle u= x^2$ so, by the chain rule, $\displaystyle \frac{df}{dx}= \frac{df}{du}\frac{du}{dx}$$\displaystyle = g(x^2)(2x)$.
The only "conditions" on g are that it be continuous.