Confused about proof of Leibniz Integral Rule If we set $G(x) = \int_{0}^{x} f(x,y) dy$, then
\begin{align}
\frac{G(x+d)-G(x)}d =& \frac{\int_{0}^{x+d} f(x+d,y)dy - \int_{0}^{x} f(x,y)dy}d \\
=& \frac{\int_{0}^{x}f(x+d,y)dy+\int_{x}^{x+d} - \int_{0}^{x}f(x,y)dy}d
\end{align}
By grouping, $$\int_{0}^{x}\frac{f(x+d,y)-f(x,y)}ddy + \int_{x}^{x+d}f(x+d,y)dy,$$ which leads to:
$$\int_{0}^{x}f'(x,y)dy + \frac{\int_{0}^{x+d}f(x+d,y)dy-\int_{0}^{x}f(x+d,y)dy}d.$$
Why is the second term not equal to $f(x+d,x)$? I know it isn't, I'm just trying to see where I'm going wrong in the proof.

Taking the limit $d\to0$, the second term is equal to $f(x,x)$. In fact you have shown that $$\frac{d}{dx} G(x)= \int_0^x f_x(x,y)dy + f(x,x).$$
