The general form of Leibniz’s Integral Rule with variable limits states:
Suppose satisfies the condition stated previously for the basic form of Leibniz’s Rule (LR-1, see “A Semi-Rigorous Derivation of Leibniz’s Rule“) . In addition, are defined and have continuous derivatives for Then for
on the left side of (1) is a function of .
(2) can be expressed as
Hence, by the chain rule,
It follows that