Definite Integral of Partial Derivative/Proof 1

Proof
From Leibniz Integral Rule:


 * $\displaystyle \frac \d {\d y} \int_{a \left({y}\right)}^{b \left({y}\right)} f \left({x, y}\right) \rd x = f \left({y, b \left({y}\right)}\right) \frac {\d b} {\d y} - f \left({y, a \left({y}\right)}\right) \frac {\d a} {\d y} + \int_{a \left({y}\right)}^{b \left({y}\right)} \frac {\partial} {\partial y} f \left({x, y}\right) \rd x$

where $a \left({y}\right)$, $b \left({y}\right)$ are continuously differentiable.

Setting $a \left({y}\right)$ and $b \left({y}\right)$ constant, so that $\exists a, b \in \R: a \left({y}\right) = a, b \left({y}\right) = b$:


 * $\dfrac {\d a} {\d y} = \dfrac {\d b} {\d y} = 0$

from which the result follows immediately.