# Partial derivatives of a definite integral

I have the following function which I need to evaluate and differentiate:

$$f(x,y)=\int_x^yg(t)dt$$

Firstly how do I find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ ?

Next, for the evaluation of $g(t)$ - I have to evaluate it numerically as the function is complex. However would I have to calculate the derivatives using this numerical procedure as well to be consistent with the evaluation? Or would the analytical derivatives above suffice?

• What about the fundamental theorem of calculus ? – Claude Leibovici Jul 1 '17 at 7:54
• The Claude Leibovici from SimSci days? If so, hi Claude this is Adrian from PipePhase/NetOpt days! Aah- so $\frac{\partial f}{\partial x}$ would be -$g(x)$ and $\frac{\partial f}{\partial y}$ would be $g(y)$ ? – dacfer Jul 1 '17 at 8:02
• Hi Adrian ! Yes, I am the one. Good to meet you again. What are you doing ? – Claude Leibovici Jul 1 '17 at 8:04
• will pm you - is my interpretation of the FTOC correct above? – dacfer Jul 1 '17 at 8:07
• Nice to see two friends meeting on MSE. Btw your interpretation of FTOC is correct. – Paramanand Singh Jul 1 '17 at 8:18