Leibniz integral rule ($\frac{d}{dx}{\int }_{a(x)}^{b(x)}f(x,t)dt=f(x,b)b^{\prime }-f(x,a)a^{\prime }+{\int }_a^b{f}_x(x,t)dt$)

For differentiable endpoints $a,b$ and a function $f$ with suitable continuous partial derivative in the parameter, the moving-endpoint rule isTo prove it locally, put $H(x,u)={\int }_c^{u}f(x,t)dt$ for a fixed reference endpoint $c$. The original integral is $H(x,b(x))-H(x,a(x))$. Apply the chain rule, use the fundamental theorem of calculus for $H_u=f$, and use differentiation under the integral sign for $H_x$. The two boundary terms account for the moving endpoints, while the remaining integral accounts for variation of the integrand with the parameter. For improper integrals, convergence and a suitable domination condition must be verified before interchanging the derivative and integral.