Definite Integral of Partial Derivative

Theorem
If $$\displaystyle f(x, y)$$ and $$\frac{\partial f}{\partial x}(x, y)$$ are continuous functions of x and y on $$D = [x_1, x_2]\times [a, b]$$, then $$\frac{\mathrm d}{\mathrm d x} \int_a^b f(x, y)\mathrm dy = \int_a^b \frac{\partial f}{\partial x}(x, y) \mathrm dy$$ for $$x\in[x_1, x_2]$$.

Proof
Define $$G(x) = \int_a^b f(x, y)\mathrm dy$$. (The continuity of f ensures that G exists.) Then by linearity of the integral, $$\frac{\Delta G}{\Delta x} = \frac{G(x + \Delta x) - G(x)}{\Delta x} = \int_a^b \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x}\mathrm dy$$. We want to find the limit of this quantity as $$\Delta x$$ approaches zero.

For each $$y\in [a, b]$$, we can consider $$\displaystyle f_y(x) = f(x, y)$$ as a separate function of the single variable $$x$$, with $$\frac{\mathrm df_y}{\mathrm dx} = \frac{\partial f}{\partial x}$$. Thus by the Mean Value Theorem, there is a number $$c_y\in (x, x + \Delta x)$$ such that $$f_y(x + \Delta x) - f_y(x) = \frac{\mathrm df_y}{\mathrm dx}(c_y)\Delta x$$, i.e., $$f(x + \Delta x, y) - f(x, y) = \frac{\partial f}{\partial x}(c_y, y)\Delta x$$.

Therefore $$\frac{\Delta G}{\Delta x} = \int_a^b \frac{\partial f}{\partial x}(c_y, y)\mathrm dy$$.

Now, pick any $$\epsilon > 0\ $$ and set $$\epsilon_0 = \frac{\epsilon}{b - a}$$. Since $$\frac{\partial f}{\partial x}$$ is continuous on the compact set D, it is uniformly continuous on D; hence for each x and y, $$\Big|\frac{\partial f}{\partial x}(x + h, y) - \frac{\partial f}{\partial x}(x, y)\Big| < \epsilon_0$$ whenever $$h$$ is sufficiently small. And since $$x < c_y < x + \Delta x\ $$, it follows that for sufficiently small $$\Delta x\ $$ that $$\Big|\frac{\partial f}{\partial x}(c_y, y) - \frac{\partial f}{\partial x}(x, y)\Big| < \epsilon_0$$ regardless of our choice of y. So we can say

$$ $$ $$ $$

But since $$\epsilon\ $$ was arbitrary, it follows that $$\lim_{\Delta x \rightarrow 0} \frac{\Delta G}{\Delta x} = \int_a^b \frac{\partial f}{\partial x}(x, y)\mathrm dy$$ and the theorem is proved.