Leibniz's Rule Proof With Definition of a Derivative

In summary: Then divide by \Delta x and take the limit as \Delta x \rightarrow 0. This should give you the definition of the derivative for G(x). Then use the chain rule to simplify the expression for \frac{dG}{dx}. As for the second part, consider the area under the curve of the integrand from x = a to x = a + \Delta x, and how this relates to the value of the integral from x = a to x = a + \Delta x. As \Delta x \rightarrow 0, what happens to this area and the value of the integral? In summary, to show Leibniz's rule, use the definition of the derivative and the chain rule. To understand the second
  • #1
PhysicsIzHard
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Homework Statement



Use the definition of the derivative to show that if G(x)=[itex]\int[/itex][itex]^{u(x)}_{a}[/itex]f(z)dz, then [itex]\frac{dG}{dx}[/itex]=f(u(x))[itex]\frac{du}{dx}[/itex]. 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[itex]\underline{x\rightarrow0}[/itex][itex]\int[/itex][itex]^{a+x}_{a}[/itex]f(z)dz=lim[itex]\underline{x\rightarrow0}[/itex]f(a)[itex]\int[/itex][itex]^{a+x}_{a}[/itex]dz. A relation like this will probably be useful in your solution to this problem.


Homework Equations



http://upload.wikimedia.org/math/4/2/c/42cf4f4861ae1266b13104c4115e7b5d.png

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?
 
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  • #2
Consider introducing the function h(x) defined by

[tex]h(x) = \int_{a}^{x} f(z) dz[/tex]

Then [itex]G(x) = h(u(x))[/itex]. What happens if you use the chain rule to differentiate this?
 
  • #3
PhysicsIzHard said:

Homework Statement



Use the definition of the derivative to show that if G(x)=[itex]\int[/itex][itex]^{u(x)}_{a}[/itex]f(z)dz, then [itex]\frac{dG}{dx}[/itex]=f(u(x))[itex]\frac{du}{dx}[/itex]. 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[itex]\underline{x\rightarrow0}[/itex][itex]\int[/itex][itex]^{a+x}_{a}[/itex]f(z)dz=lim[itex]\underline{x\rightarrow0}[/itex]f(a)[itex]\int[/itex][itex]^{a+x}_{a}[/itex]dz. A relation like this will probably be useful in your solution to this problem.


Homework Equations



http://upload.wikimedia.org/math/4/2/c/42cf4f4861ae1266b13104c4115e7b5d.png

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?

Try writing the expression for G (x + [itex]\Delta x[/itex]) and subtracting the expression for G(x).
 

1. What is Leibniz's Rule?

Leibniz's Rule, also known as the Product Rule, is a mathematical rule used to find the derivative of a product of two functions. It states that the derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function.

2. What is the proof for Leibniz's Rule?

The proof for Leibniz's Rule involves using the definition of a derivative and manipulating it algebraically to show that it is equivalent to the product rule. It involves using the limit definition of a derivative and the properties of limits. The full proof can be found in most calculus textbooks.

3. How is Leibniz's Rule used in calculus?

Leibniz's Rule is used in calculus to find the derivative of a product of two functions. It is often used when finding the derivative of more complex functions that cannot be easily differentiated using basic rules, such as the power rule or chain rule.

4. Can Leibniz's Rule be extended to more than two functions?

Yes, Leibniz's Rule can be extended to more than two functions. This is known as the General Leibniz Rule and states that the derivative of a product of n functions is equal to the sum of all possible products of the derivatives of the individual functions. However, this can become increasingly complicated as the number of functions increases.

5. What are some real-world applications of Leibniz's Rule?

Leibniz's Rule has many real-world applications, particularly in physics and engineering. It is used to find the rate of change of quantities that are dependent on multiple changing variables, such as velocity and acceleration. It is also used in economics, biology, and other fields to model and analyze relationships between different variables.

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