# 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

Staff Emeritus
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 $\frac {d}{dx} \int_0^x f(t)dt =f(x)$