Graduate Multi-variable function depending on the Heaviside function

[CCMarie]

How can I calculate ∂/∂t(∫₀¹ f(x,t,H(x-t)*a)dt), where a is a constant, H(x) is the Heaviside step function, and f is

I know it must have something to do with distributions and the derivative of the Heaviside function which is ∂/∂t(H(t))=δ(x)... but I don't understand how can I work with the Heaviside function being an argument of the function...

 

[RPinPA]

And f is...?

Anyway, I don't think it matters. $\int_0^1 f(...) dt$ does not actually depend on $t$. Once you do the integration with respect to $t$, $t$ no longer appears as a variable. So the derivative is 0.