Differentiation Rule of Leibnitz

Suppose that

$$F_z(z) = \int\limits_{a(z)}^{b(z)} f(x,z) dx.$$

Then,

$$f_z(z) = \frac{dF_z(z)}{dz} = \frac{db(z)}{dz}f(b(z),z)-\frac{da(z)}{dz}f(a(z),z)+\int\limits_{a(z)}^{b(z)}\frac{\partial f(x,z)}{\partial z} dx$$

See page 181 of Papoulis.