While trying to solve a heat conduction problem, I stumbled upon a kind of equation which, in general form, can be written as:
\begin{equation} f(t) = \int_0^t \frac{g(x)}{\sqrt{\tau(t)-\tau(x)}}dx \end{equation}
In the problem I am considering, this equation is coupled to others where the derivative $\frac{\partial f}{\partial t}$ appears. Therefore I tried to evaluate such derivative using Leibniz's integral rule (see link). The result I obtained is:
\begin{equation} \frac{\partial f(t)}{\partial t} = \frac{g(t)}{\sqrt{\tau(t)-\tau(t)}}-\frac{1}{2}\frac{\partial \tau(t)}{\partial t}\int_0^t\frac{g(x)}{(\tau(t)-\tau(x))\sqrt{\tau(t)-\tau(x)}}dx \end{equation} The integrand on the RHS still has a vertical asymptote at $x=t$, but I am not too worried about it (it is likely the integral converges).
However, the first terms is infinite, which make sense since it is just the evaluation of the integrand in $f(t)$ at $x=t$, i.e. where the function does not exist.
I must assume that Leibniz's integral rule is applicable only if the integrand is continuous in the entire interval of integration, including the boundaries. This would mean that I cannot perform the derivative with wrt $t$ of $f(t)$ by standard methods.
I was wondering if a derivative of such kind of equation is just mathematically unfeasible or if there are more sophisticated techniques to approach the problem.
Cheers.