Does this equality always hold? Is it true in general that $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x} \int_0^{x} f(u,x) \mathrm{d}u = \int_0^{x} \left( \frac{\mathrm{d}}{\mathrm{d}x} f(u,x) \right)\mathrm{d}u +f(x,x )$ ?
Thank you for your help!

In short, it is true if $f$ and the partial derivative with respect to $f$ are continuous in the region of differentiation.
