= Solution
Differentiate the rewritten expression. Its new kernel vanishes at the moving upper endpoint, and its <partial derivative> is an <integrable> inverse square root. The <derivative of an Abel integral> is
$$
\boxed{g'(x)=\frac{h(0)}{\sqrt x}+\int_0^x\frac{h'(t)}{\sqrt{x-t}}\,dt,\qquad x>0.}
$$
For a rigorous route, rewrite $2\int_0^xh'(t)\sqrt{x-t}\,dt$ as $\int_0^x\int_0^s h'(t)/\sqrt{s-t}\,dt\,ds$ using <Fubini's theorem>, and apply the <fundamental theorem of calculus> to the continuous inner <integral>. The assumption $h(0)\ne0$ implies $g(x)=2h(0)\sqrt x+O(x^{3/2})$, so the extension $g(0)=0$ has no finite right <derivative> at zero.
Back to article page
