# Derivative of definite integral?

• Sep 16th 2009, 10:04 PM
scorpion007
Derivative of definite integral?
Is it possible to take the derivative of a definite integral? I guess if we have an indefinite one, like:
$\int f(x)\,dx$, taking its derivative, we would get $f(x)$, right?

What if we had $\int_a^b f(x)\,dx$?
• Sep 16th 2009, 10:16 PM
seld
wouldn't it be 0? because the solution to the indefinite integral would be a constant, derivative of a constant = 0.
• Sep 17th 2009, 05:59 AM
HallsofIvy
Yes, that is correct. $\int_a^b f(x) dx$ is a constant and its derivative is 0.
• Sep 18th 2009, 06:34 PM
scorpion007
Thanks guys.

seld, you meant definite integral right? (not indefinite).
• Sep 18th 2009, 06:43 PM
VonNemo19
Quote:

Originally Posted by scorpion007
Thanks guys.

seld, you meant definite integral right? (not indefinite).

Yes, because indefinte integrals can be written as a function of x provided that one of the limits of integration is constant and the other is some x. But a definite integral is a number, and as hallsofivy said, the derivative of a number is always zero.
• Sep 20th 2009, 11:32 AM
seld
Quote:

Originally Posted by scorpion007
Thanks guys.

seld, you meant definite integral right? (not indefinite).

er . . . yes, wow maybe I should stop posting after explaining analysis for about 2 hours to people . . . yes that's a typo thingy
• Jun 8th 2010, 02:24 PM
Plato
Quote:

Originally Posted by scorpion007
Is it possible to take the derivative of a definite integral? I guess if we have an indefinite one, like:
$\int f(x)\,dx$, taking its derivative, we would get $f(x)$, right?

What if we had $\int_a^b f(x)\,dx$?

$\int_a^b f(x)\,dx$ is a definite integral.
A definite integral is a number. What is the derivative of a number?