Question about derivative of an integral

1. Apr 16, 2015

Mr Davis 97

What does it mean to say that $\displaystyle\frac{d }{d x}\int f(x)dx = f(x)$? Does this somehow relate to the fundamental theorem of calculus? If so, how?

2. Apr 16, 2015

Raghav Gupta

What is the fundamental theorem according to you?

3. Apr 16, 2015

Staff: Mentor

It means that if you find $$F(x) = \int {f(x)dx}$$ (i.e. the antiderivative, a.k.a. indefinite integral) and then you find $$\frac{d}{dx}F(x)$$ you get $f(x)$ back again.

4. Apr 17, 2015

Staff: Mentor

Another way to look at it is that differentiation and antidifferentiation are inverse operations.

5. Apr 17, 2015

HallsofIvy

The "Fundamental Theorem of Calculus" has two parts:
1) If we define $F(x)= \int_a^x f(x)dx$ then $dF/dx= f(x)$.
2) If f(x)= dF/dx then $F(x)= \int f(x) dx+$ some constant.

6. Apr 19, 2015

Mr Davis 97

How are those two parts different? Also, isn't there a part that relates the definite integral to the antiderivative?

7. Apr 20, 2015

SteamKing

Staff Emeritus
In 1) above, F(x) is the antiderivative of f(x).

The First Fundamental Theorem of Calculus is more often expressed as:

$\int_a^b f(x)dx = F(b) - F(a)$, where F(x) is the antiderivative of f(x), as defined in part 2) above.

8. Apr 20, 2015

WWGD

Actually, the FTC says that $\frac {d}{dx} \int_0^x f(t)dt =f(x)$