Thread: Integral and derivate

1. Integral and derivate

We know that f(x)=
x^2
∫g(t)dt
0

What is f'(x) and which conditions are necessary for g(t)?

2. Originally Posted by antero
We know that f(x)=
x^2
∫g(t)dt
0

What is f'(x) and which conditions are necessary for g(t)?
Let $u = x^2$. Then $y = \int_0^u g(t) \, dt$.

From the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = g(u) \cdot 2x = 2x g(x^2)$.

3. Originally Posted by antero
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 $\int_a^u g(t) dt$, with respect to u, is g(u). Here, $u= x^2$ so, by the chain rule, $\frac{df}{dx}= \frac{df}{du}\frac{du}{dx}$ $= g(x^2)(2x)$.

The only "conditions" on g are that it be continuous.