Leibniz integral rule
/ LYBE-nitz /
The Leibniz integral rule answers a precise question: if an integral has a parameter both in its integrand and in its limits of integration, how does the whole thing change as the parameter moves? It is the complete formula for differentiating I(t) = integral from a(t) to b(t) of f(x, t) dx, and it stitches together two effects — the integrand shifting, and the endpoints sliding.
The rule has three pieces. The derivative of I(t) equals: first, f evaluated at the upper limit times the rate the upper limit moves, db/dt (the curve gains area as its right edge advances); minus f evaluated at the lower limit times da/dt (it loses area as the left edge advances); plus the integral over the interval of the partial derivative of f with respect to t (the integrand itself changing everywhere inside). The two boundary terms are exactly the chain-rule contributions of the moving limits; the integral term is plain differentiation under the integral sign. When the limits are constants, the boundary terms vanish and only that interior term survives.
This is the workhorse behind moving-boundary problems, transport theorems in fluid mechanics (the Reynolds transport theorem is a multidimensional Leibniz rule for a region carried by a flow), and any rate-of-change of an accumulated quantity over a growing window. A common slip is to remember only the integral term and forget the two boundary contributions when the limits actually depend on t — that omission quietly drops real physics, like the flux through a moving surface.
d/dt of the integral from 0 to t^2 of cos(t x) dx equals cos(t * t^2) * (2t) + integral from 0 to t^2 of -x sin(t x) dx — boundary term plus the interior parameter-derivative.
The moving upper limit t^2 contributes a boundary term; the parameter t inside cos contributes the integral term.
When only the limits depend on the parameter and the integrand does not, the rule collapses to the second Fundamental Theorem of Calculus with the chain rule. The full rule is needed precisely when both vary at once.