Differentiating an Integral of $f(x,t)$

Question

How would I do the following calculation?

$$ \frac{d}{dx}( \int_0^x{(x-t)f^{''}(t)dt}) $$

I tried it and I got $f^{'}(x)$, but I don't think I did it correctly.

Answer 1

Here is a start.

$$ \frac{d}{dx}\left( \int_0^x{(x-t)f^{''}(t)dt}\right) = \frac{d}{dx} x \int_0^x f^{''}(t)dt - \frac{d}{dx} \int_0^x tf^{''}(t)dt=\dots\,. $$

Can you finish it?

Added: write

$$ g(x) = \int_0^x f^{''}(t)dt $$

in the above equation and then you need to use the product rule.

Note:

$$ \frac{d}{dx} \int_a^x h(t) dt = h(x).$$