# Leibniz's Rule Proof With Definition of a Derivative

## Homework Statement

Use the definition of the derivative to show that if G(x)=$\int$$^{u(x)}_{a}$f(z)dz, then $\frac{dG}{dx}$=f(u(x))$\frac{du}{dx}$. This is called Leibniz's rule.

Also, by thinking of the value of an integral as the area under the curve of the integrand (and drawing a picture of that area), convince yourself that the following is true: lim$\underline{x\rightarrow0}$$\int$$^{a+x}_{a}$f(z)dz=lim$\underline{x\rightarrow0}$f(a)$\int$$^{a+x}_{a}$dz. A relation like this will probably be useful in your solution to this problem.

## The Attempt at a Solution

I have tried to sub G(x) into the definition of the derivative equation but that gets me no where. Any ideas anyone on where to start this?

jbunniii
Homework Helper
Gold Member
Consider introducing the function h(x) defined by

$$h(x) = \int_{a}^{x} f(z) dz$$

Then $G(x) = h(u(x))$. What happens if you use the chain rule to differentiate this?

Chestermiller
Mentor

## Homework Statement

Use the definition of the derivative to show that if G(x)=$\int$$^{u(x)}_{a}$f(z)dz, then $\frac{dG}{dx}$=f(u(x))$\frac{du}{dx}$. This is called Leibniz's rule.

Also, by thinking of the value of an integral as the area under the curve of the integrand (and drawing a picture of that area), convince yourself that the following is true: lim$\underline{x\rightarrow0}$$\int$$^{a+x}_{a}$f(z)dz=lim$\underline{x\rightarrow0}$f(a)$\int$$^{a+x}_{a}$dz. A relation like this will probably be useful in your solution to this problem.

## Homework Equations

Try writing the expression for G (x + $\Delta x$) and subtracting the expression for G(x).