How can I prove that $$\frac{d}{dz} \left( \int_C f (z, w) dw\right) =\int_C\frac { \partial f }{\partial z} (z, w) dw$$ where $C $ is a smooth curve in $\mathbb C $ and $f$ is a function of two complex variables with the appropriate smoothness conditions ?
The formula seems to be valid, but I don't know how to prove it, since in the real case, we used the mean value theorem. Help would be very much appreciated.