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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

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

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