# Does this work to take derivatives of integrals?

I'm working on a homework problem where I need to do something that looks like this (this isn't the exact question - I just want to check my reasoning for my approach). Given:

$$f(t)=\int_a^b{g(x,t)dx}$$

I need to find $$df/dt$$.

The approach I've come up with is:

$$\frac{df}{dt}=\lim_{h\to0}\frac{\int_a^b{g(x,t+h)dx}- \int_a^b{g(x,t)dx}}{h}=\lim_{h\to0}\int_a^b{\frac{g(x,t+h)-g(x,t)}{h}}dx=\int_a^b{\lim_{h\to0}\frac{g(x,t+h)-g(x,t)}{h}}dx=\int_a^b{\frac{dg}{dt}dx}$$

In other words, to calculate $$df/dt$$, I can calculate $$dg/dt$$, and then integrate that with respect to $$x$$ from $$a$$ to $$b$$.

Is this correct? It feels right to me, but I'm not sure about bringing the limit inside the integral.

Apologies if the notation is bad. I wasn't sure how to: (a) make bigger integrals; (b) do multiple lines with the equal signs lining up; or (c) get the limit arrow part under the limit.

## 2 Answers

"Liebnitz rule", that zkutch refers to, says, in general, that $$\frac{d}{dt}\int_{\alpha(t)}^{\beta(t)} f(x,t) dx= f(\beta(t),t)- f(\alpha(t),t)+ \int_{\alpha(t)}^{\beta(t)} \frac{\partial f}{\partial t} dx$$.

In book John M.H. Olmsted - Advanced calculus-Prentice Hall (1961), on pages 321-324 you can find different variants on how to differentiate under integral sign (Lebnitz's rule). One theorem asserts:

If $$f(x,y)$$ and $$f'_x(x,y)$$ are continuous on closed rectangle $$a \leqslant x \leqslant b, c \leqslant y \leqslant d$$, then function $$F(x)=\int\limits_{c}^{d}f(x,y)dy$$ is differentiable for $$a \leqslant x \leqslant b$$ and

$$F'(x)=\int\limits_{c}^{d}f'_x(x,y)dy$$

You can find there generalizations for case with continuous functions for integral bounds and improper integral.

• Lucky Olmsted! I read his book. Good one! + – Mikasa Apr 23 at 13:17
• He has also another excellent book "Counterexamples in Analysis". But why "lucky", @Mikasa? Some story to know? – zkutch Apr 23 at 14:00
• Cause there are a few books which did what he did with a bit different approach. Like Adams. I read read the book due to teaching Calculus I – Mikasa Apr 23 at 14:43