Leibniz's Rule

126. Leibniz's Rule

126.1. Leibniz’s Rule

Differentiation under the integral sign. When the integrand and the limits depend on a parameter $𝑄$ , the derivative splits into three pieces: one from the integrand changing, two from the moving limits.

\frac{d}{d 𝑄} \int_{𝑎 (𝑄)}^{𝑏 (𝑄)} ℎ (𝑥, 𝑄) d 𝑥 = \underset{integrand varies}{\underset{⏟}{\int_{𝑎 (𝑄)}^{𝑏 (𝑄)} \frac{𝜕 ℎ}{𝜕 𝑄} d 𝑥}} + \underset{upper limit moves}{\underset{⏟}{ℎ (𝑏 (𝑄), 𝑄) \cdot 𝑏^{'} (𝑄)}} - \underset{lower limit moves}{\underset{⏟}{ℎ (𝑎 (𝑄), 𝑄) \cdot 𝑎^{'} (𝑄)}}

Reading the three terms:

Integrand term — freeze the limits, differentiate inside.
Upper-limit term — rate the top edge sweeps out new area: $ℎ$ at $𝑥 = 𝑏 (𝑄)$ times $𝑏^{'} (𝑄)$ .
Lower-limit term — area lost at the bottom edge, subtracted.

126.1.1. Special cases

Fixed limits ( $𝑎, 𝑏$ constant) — only the integrand term survives:

\frac{d}{d 𝑄} \int_{𝑎}^{𝑏} ℎ (𝑥, 𝑄) d 𝑥 = \int_{𝑎}^{𝑏} \frac{𝜕 ℎ}{𝜕 𝑄} d 𝑥

Constant integrand, moving upper limit (the fundamental theorem):

\frac{d}{d 𝑄} \int_{𝑎}^{𝑄} ℎ (𝑥) d 𝑥 = ℎ (𝑄)

Example

The newsvendor overage term $\int_{0}^{𝑄} (𝑄 - 𝑥) 𝑓 (𝑥) d 𝑥$ has $ℎ (𝑥, 𝑄) = (𝑄 - 𝑥) 𝑓 (𝑥)$ , upper limit $𝑏 (𝑄) = 𝑄$ , lower limit fixed:

\frac{d}{d 𝑄} \int_{0}^{𝑄} (𝑄 - 𝑥) 𝑓 (𝑥) d 𝑥 = \int_{0}^{𝑄} 𝑓 (𝑥) d 𝑥 + \underset{= 0}{\underset{⏟}{(𝑄 - 𝑄) 𝑓 (𝑄)}} = 𝐹 (𝑄)

The boundary term vanishes because the integrand is zero at $𝑥 = 𝑄$ .