Seemingly Simple Derivative (as a limit) Problem

In summary, the conversation discusses a problem involving the limit of a function and its derivative. The individuals are trying to find a way to show that lim [f(ax)-f(bx)]/x = f'(0)(a-b) as x approaches 0. They suggest using the definition of the derivative and rearranging the terms, but also consider using the chain rule.
  • #1
luke8ball
22
0
I'm having trouble showing the following:

lim [f(ax)-f(bx)]/x = f'(0)(a-b)
x→0

I feel like this should be really easy, but am I missing something? I tried to use the definition of the derivative, but I know I can't just say f(ax)-f(bx) = (a-b)f(x).

Any ideas?
 
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  • #2
Try to add zero in your numerator in the shape f(0)-f(0), and see if you can rearrange it in a clever manner.
 
  • #3
You mean so that I get:

[lim f(ax) - f(0)]/x - [lim f(bx) - f(0)]/x
x→0 x→0

I had thought about that, but I still don't see how that gives me af'(0) - bf'(0)...
 
  • #4
Think chain rule..
 
  • #5
A further hint:
Let g(x)=ax. Then, g(0)=0
 

FAQ: Seemingly Simple Derivative (as a limit) Problem

What is a seemingly simple derivative (as a limit) problem?

A seemingly simple derivative (as a limit) problem is a type of calculus problem that involves finding the instantaneous rate of change of a function at a specific point. The problem may appear simple at first glance, but it requires the use of limits and other advanced calculus techniques to solve.

How do you solve a seemingly simple derivative (as a limit) problem?

To solve a seemingly simple derivative (as a limit) problem, you first need to identify the function and the point at which you want to find the derivative. Then, you can use the definition of a derivative, which involves taking the limit as the change in the input approaches zero. This limit will give you the slope of the tangent line at the given point, which is the value of the derivative.

What are the common mistakes made when solving a seemingly simple derivative (as a limit) problem?

One common mistake is forgetting to take the limit as the change in the input approaches zero. Another mistake is not using the correct formula for finding the derivative, such as using the power rule instead of the definition of a derivative. It is also important to carefully evaluate the limit, as sometimes it can be indeterminate and require additional algebraic manipulation.

Why are seemingly simple derivative (as a limit) problems important in math and science?

Seemingly simple derivative (as a limit) problems are important in math and science because they allow us to calculate instantaneous rates of change, which is a fundamental concept in calculus. This is useful in many applications, such as physics, economics, and engineering, where it is necessary to understand how a quantity changes at a specific point in time.

What are some real-world examples of seemingly simple derivative (as a limit) problems?

Real-world examples of seemingly simple derivative (as a limit) problems include finding the velocity of a moving object at a specific point in time, calculating the growth rate of a population, and determining the rate of change of a stock price. These problems all involve finding the instantaneous rate of change of a function at a given point, which can be solved using the definition of a derivative.

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