What ins the derivative of \int_0^t\varphi(t,u)du

Question

Let $f(t)=\int_0^t\varphi(t,u)du$ where $\varphi$ is a continuos function. Do yo know how to compute the derivative of f? It is not possible to directly apply the fundamental theorem of calculus.

Answer 1

I'm assuming you meant the derivative of $f$ in your post.

In a simple setting we need some assumptions on $\varphi$ namely that $\dfrac{\partial\varphi}{\partial t}$ exists and is continuous (to simplify things). Then the derivative of $f$ is obtained by $$f'(t)=\int_0^t \dfrac{\partial\varphi}{\partial t}(t,u)\,du+\varphi(t,t)$$ We get two bits, in the first one we differentiate under the integral sign while the second looks like an application of the fundamental theorem of calculus.

Note that the integral above is well-defined since $\dfrac{\partial\varphi}{\partial t}$ is continuous.