= Leibniz integral rule
{c}
{title2=$\frac d{dx}\int_{a(x)}^{b(x)}f(x,t)dt=f(x,b)b'-f(x,a)a'+\int_a^bf_x(x,t)dt$}
{wiki}
For differentiable endpoints $a,b$ and a <function> $f$ with suitable continuous <partial derivative> in the parameter, the moving-endpoint rule is
$$
\boxed{\frac d{dx}\int_{a(x)}^{b(x)}f(x,t)\,dt=f(x,b(x))b'(x)-f(x,a(x))a'(x)+\int_{a(x)}^{b(x)}f_x(x,t)\,dt.}
$$
To 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>.
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