Definite Integral of Partial Derivative

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

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

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

Proof
Define $$G \left({x}\right) = \int_a^b f \left({x, y}\right) \mathrm dy$$.

The continuity of $$f$$ ensures that $$G$$ exists.

Then by linearity of the integral:
 * $$\frac{\Delta G}{\Delta x} = \frac{G \left({x + \Delta x}\right) - G \left({x}\right)} {\Delta x} = \int_a^b \frac{f \left({x + \Delta x, y}\right) - f \left({x, y}\right)} {\Delta x}\mathrm dy$$

We want to find the limit of this quantity as $$\Delta x$$ approaches zero.

For each $$y \in \left[{{a} \,. \, . \, {b}}\right]$$, we can consider $$f_y \left({x}\right) = f \left({x, y}\right)$$ 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 $$c_y \in \left({x, x + \Delta x}\right)$$ such that $$f_y \left({x + \Delta x}\right) - f_y \left({x}\right) = \frac {\mathrm df_y} {\mathrm dx} \left({c_y}\right) \Delta x$$.

That is:
 * $$f(x + \Delta x, y) - f(x, y) = \frac{\partial f}{\partial x}(c_y, y)\Delta x$$

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

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 $$y$$:
 * $$\left|{\frac {\partial f} {\partial x} \left({x + h, y}\right) - \frac {\partial f} {\partial x} \left({x, y}\right)}\right| < \epsilon_0$$

whenever $$h$$ is sufficiently small.

And since $$x < c_y < x + \Delta x\ $$, it follows that for sufficiently small $$\Delta x\ $$ that:
 * $$\left|{\frac {\partial f} {\partial x} \left({c_y, y}\right) - \frac {\partial f} {\partial x} \left({x, y}\right)}\right| < \epsilon_0$$

regardless of our choice of $$y$$.

So we can say:

$$ $$ $$ $$

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