# Differentiate an integral

## Homework Statement

Hi

Say I have for example

$$f(x) = \int_0^x {e^{ - (x - x')} g(x')\,dx'}$$

Then is it correct that the derivative of f(x), f'(x), is given by

$$f'(x) = g'(x) - g(x)$$

obtained by differentiating the integrand, and evaluate the result at x'=x?

Best.

HallsofIvy
Homework Helper
No, it is not.

Laplace's formula:
$$\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x,y)dy= f(x, \beta(x))\frac{d\beta}{dx}- f(x,\alpha(x))\frac{d\alpha}{dx}+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f}{\partial x} dy$$

In your problem, $\beta(x)= x$, $\alpha(x)= 0$, and $f(x,x')= e^{x-x'}g(x')$.

Pretty impressive that you managed to read and reply in ~3 minutes. A tip of the hat to you!

Is that the most general way? Or does it all come down to the fundemental theorem of calculus?

hello,

that is not quite correct. the fundamental theorem of calculus tells us that if

$$f(x) = \int_0^x{g(t)}dt$$

then $$f'(x) = g(x)$$ .

but your function is of the form:

$$f(x) = \int_0^x{h(x,t) g(t)}dt$$

which is more complicated.

try integration by parts on the right hand side before differentiating.

hope this helps.

I see, thanks to both of you.

Best.

cheers niles,

p.s. i believe if you try my method you will see the solution falls out very nicely and quickly.