# Leibniz's Rule Proof With Definition of a Derivative

PhysicsIzHard

## 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?

## Answers and Replies

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?

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).