Solution
= Solution
The <Leibniz integral rule> combines contributions from both moving endpoints with <differentiation under the integral sign>:
$$
\boxed{\frac d{dx}\int_{a(x)}^{b(x)}f(x,t)\,dt=f(x,b(x))b'(x)-f(x,a(x))a'(x)+\int_{a(x)}^{b(x)}\frac{\partial f}{\partial x}(x,t)\,dt.}
$$
The <partial derivative> in the last <integral> holds $t$ fixed. Suitable regularity and convergence are assumed as stipulated.