# Thread: derivative of a definite integral

1. ## derivative of a definite integral

is this done by cos(e^3)-cos(e^1)

2. ## Re: derivative of a definite integral

$\displaystyle \int_1^3\cos(e^y)\,dy$ is a constant; it does not depend on x. Therefore, its derivative with respect to x is zero.

3. ## Re: derivative of a definite integral

The answer emakarov gave is correct, but I'll bet you copied the question incorrectly. Either that or you got a trick question. Did you perhaps mean $\displaystyle \frac{d}{dx}\int_1^x\cos{e^y}dy$? By the Fundamental Theorem of Calculus, it is equal to $\displaystyle \cos{e^x}$.

- Hollywood

4. ## Re: derivative of a definite integral

it was a trick question, thanks!

what if it were defined on from 2 to x would we still have cos e^x as the derivative of the integral?

MH

5. ## Re: derivative of a definite integral

Originally Posted by kingsolomonsgrave
what if it were defined on from 2 to x would we still have cos e^x as the derivative of the integral?
Yes.