# How Can Taking The Derivative Of A Definite Integral Produce A Sum of A Term Similar To The Integrand and Another Integral With A Similar Integrand

On page 138 In the book "Elementry Differnetial Equations With Boundary Value Problems" 4th Eddition by William R. Derrick and Stanley I. Grossman, the authors take the derivative of a definite integral (The first equation with a red line beside it) and and produce an equation (the second equation with a red line beside it) that is the sum of a term that looks like the integrand and another definite integral.

How did the authors do this?

Page 138 From The Book

This follows from the Leibniz rule in 1-D: $$\frac{d}{dx}\int_{a(x)}^{b(x)}\phi(x,t)dt=\phi(x,b(x))b'(x)-\phi(x,a(x))a'(x)+\int_{a(x)}^{b(x)}\frac{\partial}{\partial x}\phi(x,t)dt$$