# Thread: differentiating under the integral sign

1. ## differentiating under the integral sign

To prove that $\frac {d}{dt} \int^{b}_{a} f(x,t) \ dx = \int^{b}_{a} \frac{d}{dt} f(x,t) \ dx$ at $t=t_{0}$

is it sufficient to show that $f(x,t)$ and $\frac{d}{dt}f(x,t)$ are continous for $x$ in the range of integration and for $t$ in a interval about $t_{0}$? And if you want to differentiate under the integral sign multiple times, is it sufficient to show that each derivative of $f(x,y)$ is continous for $x$ in the range of intergration and for $t$ in a interval about $t_{0}$?

And what about improper integral $\frac {d}{dt} \lim_{b \to \infty} \int^{b}_{a} f(x,t) \ dx$ ?

Is it sufficient to show in addition that $|f(x,t)|$ and $\Big|\frac{d}{dt}f(x,t)\Big|$ are bounded above by functions independent of $t$ and the integrals of both of those functions converge over the interval [a, $\infty)$?

2. I must ask the worst questions.

3. Hello,

Leibniz integral rule - Wikipedia, the free encyclopedia

Is it what you're looking for ? (link to the proof in wikipedia)

4. Originally Posted by Moo
Hello,

Leibniz integral rule - Wikipedia, the free encyclopedia

Is it what you're looking for ? (link to the proof in wikipedia)
Yes. Thanks.

But I'm still uncertain about improper integrals.