Leibniz integral rule?

Suppose that $a$ and $b$ are fixed constants and let $f(x) = \int_{x'=a}^{x'=b} g(x,x') \, dx'$. If x' is in the same direction as x, why is it true that

$$\frac{df}{dx} = g(x,x') \Big|_{x'=a}^{x'=b}?$$

If $g=g(x+x')$, then
\begin{align}f'(x) &= \frac{d}{dx}\int_a^b g(x+x') dx'\\\\ &=\int_a^b \frac{dg(x+x') }{dx}dx'\\\\ &=\int_a^b \frac{dg(x+x') }{dx'}dx'\\\\ &=g(x+b)-g(x+a) \end{align}