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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 isFor a rigorous route, rewrite $2{\int }_0^x{h}^{\prime }(t)\sqrt{x-t}dt$ as ${\int }_0^x{\int }_0^s{h}^{\prime }(t)/\sqrt{s-t}dtds$ using Fubini's theorem, and apply the fundamental theorem of calculus to the continuous inner integral. The assumption $h(0)\ne 0$ implies $g(x)=2h(0)\sqrt{x}+O(x^{3/2})$, so the extension $g(0)=0$ has no finite right derivative at zero.

The kernel has an integrable singularity at the moving endpoint $t=x$. Direct substitution into the ordinary Leibniz integral rule would produce the undefined boundary value $h(x)/\sqrt{x-x}$ and the nonintegrable differentiated kernel $-h(t)/[2(x-t{)}^{3/2}]$. Those separately divergent expressions cannot be treated as ordinary finite terms. The derivative of an Abel integral is obtained by rewriting the integral before differentiating.