# Fundamental theorem of calculus

$\frac{d}{dx} \int_a^b f(x) dx=f(b)$

This is something I can churn through mechanically but I never "got." Any links / explanations that can help build my intuition about this would be helpful.

## Answers and Replies

Mark44
Mentor
$\frac{d}{dx} \int_a^b f(x)=f(b)$

This is something I can churn through mechanically but I never "got." Any links / explanations that can help build my intuition about this would be helpful.

What you have is incorrect, assuming that both a and b are constants.
$$\frac{d}{dx} \int_a^b f(x) dx =0$$

The way this is usually presented is like so:

$$\frac{d}{dx} \int_a^x f(t) dt =f(x)$$

Mark44
Mentor
For an explanation, let's assume that F(x) is an antiderivative of f(x). IOW, F'(x) = f(x).
Then
$$\int_a^x f(t) dt = F(x) - F(a)$$
So $$d/dx \int_a^x f(t) dt = d/dx( F(x) - F(a)) = F'(x) - 0 = f(x)$$