# Finding a derivative through the limit method

• lyarbrough
In summary, the problem involves finding the limit of [(2+h)^5 -2^5]/h as h approaches 0. The student already has the solution, but wants to make sure their understanding is correct. They are supposed to use the definition of the derivative to solve this problem. The solution involves substituting 2 in for f(x) and then using the chain rule to find f'(x). However, the student is unsure of how f'(x) is determined and if it is just a matter of substituting x for 2. The correct answer is 80.
lyarbrough

## Homework Statement

Hi, I'm trying to understand how you solve for the problem lim [(2+h)^5 -2^5]/h as h→0
I already have the solution, but I want to make sure my understanding of how it's done is correct.

## Homework Equations

I'm suppose to be using the definition of the derivative [f(x+h)-f(x)]/h

## The Attempt at a Solution

So what I have is that lim [(2+h)^5 -2^5]/h as h→0 is = f'(2). I'm assuming that's because 2 was substituted in for f(x) in the definition of a derivative.

The next step I have is that f(x)= ? and then f'(x) = x^5. I'm just wondering if it's just re substituting in x for the 2? I know how to solve it from there with the chain rule, I'm just wondering how they determine f'(x). Thanks!

lyarbrough said:

## Homework Statement

Hi, I'm trying to understand how you solve for the problem lim [(2+h)^5 -2^5]/h as h→0
I already have the solution, but I want to make sure my understanding of how it's done is correct.

## Homework Equations

I'm suppose to be using the definition of the derivative [f(x+h)-f(x)]/h

## The Attempt at a Solution

So what I have is that lim [(2+h)^5 -2^5]/h as h→0 is = f'(2). I'm assuming that's because 2 was substituted in for f(x) in the definition of a derivative.

The next step I have is that f(x)= ? and then f'(x) = x^5. I'm just wondering if it's just re substituting in x for the 2? I know how to solve it from there with the chain rule, I'm just wondering how they determine f'(x). Thanks!

Since you say you know the answer, the limit of that is 80. That's the answer. What's the question? Are they asking you to guess what f(x) would give you that difference quotient to determine f'(2)? That shouldn't be hard. Are you sure they didn't say f(x)=x^5, not f'(x)=x^5?

Last edited:
Hi lyrabrough. Welcome to Physics Forums.

The way they want you to do this problem is not by actually taking the derivative. They want you to find the limit of that ratio as h approaches zero.

What you need to do is factor that numerator into two factors. One of the factors will cancel with the h in the denominator. The other factor will be evaluated at h = 0. Consider this:

x5-y5=(x-y)(x4+x3y+x2y2+xy3+y4)

Chet

## What is the limit method for finding a derivative?

The limit method is a mathematical technique used to find the derivative of a function at a specific point. It involves taking the limit as the change in the independent variable approaches zero.

## Why is the limit method used to find derivatives?

The limit method is used because it is a precise and accurate way to find the instantaneous rate of change of a function at a specific point. It allows us to find the slope of a curve at a single point rather than an average rate of change over an interval.

## What are the steps for finding a derivative through the limit method?

The steps for finding a derivative through the limit method are as follows:

1. Identify the function and the point at which you want to find the derivative.
2. Write out the limit expression using the definition of the derivative.
3. Simplify the expression by factoring, canceling out common terms, and using algebraic manipulations.
4. Take the limit as the change in the independent variable approaches zero.
5. Solve the limit to find the derivative at the given point.

## What are some common mistakes when using the limit method to find derivatives?

Some common mistakes when using the limit method to find derivatives include:

• Forgetting to take the limit at the end of the calculation.
• Not simplifying the expression fully before taking the limit.
• Using incorrect algebraic manipulations.
• Forgetting to write the function in terms of the independent variable.

## Can the limit method be used to find derivatives of all functions?

No, the limit method can only be used to find derivatives of continuous functions. It cannot be used for functions with discontinuities or sharp corners, as the limit does not exist at those points.

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