Definite Integral of Partial Derivative

Theorem
Let $f \left({x, y}\right)$ and $\dfrac {\partial f} {\partial x} \left({x, y}\right)$ be continuous functions of $x$ and $y$ on $D = \left[{x_1 \,.\,.\, x_2}\right] \times \left[{a \,.\,.\, b}\right]$.

Then:
 * $\displaystyle \frac {\rd} {\rd x} \int_a^b f \left({x, y}\right) \rd y = \int_a^b \frac {\partial f}{\partial x} \left({x, y}\right) \rd y$

for $x \in \left[{{x_1} \,.\,.\, {x_2}}\right]$.