4.6.2 Leibnitz's Rule

Leibnitz's Rule for differentiation of integrals

 

$$\frac{\partial }{\partial x} \int_{f(x)}^{g(x)}\,dx'\, F(x,x') F(x,g(x)) \frac{\partial g(x) }{\partial x} -F(x,f(x)) \frac{\partial f(x) }{\partial x}$$ =

$$\int_{f(x)}^{g(x)}\,dx'\, \frac{\partial }{\partial x}F(x,x')$$ + (4.65)

is employed especially when dealing with vertical integrals where the bottom topography z=-H and free surface height $z=\eta$ are integration limits.