Leibniz’s rule (LR-1) states:
Let be continuous and have a continuous derivative in a domain of plane that includes the rectangle
I will derive LR-1 semi-rigorously as follows:
Integrate (1-1) with respect to from a constant to a variable , we have
While and are functions of , is a constant.
differentiate (1-2) with respect to gives
In the three-dimensional -space, the double integral of a continuous function with two independent variables, , may be interpreted as a volume between the surface and the -plane:
We see from Fig. 1 that on one hand,
but on the other hand,
Since (2-1) and (2-2) amounts to the same thing, it must be true that
In other words, the order of integration can be interchanged.