Finding the Derivative: How to Solve for f'(x) Using the Limit Definition

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Discussion Overview

The discussion revolves around finding the derivative of the function f(x) = 3x + 2 using the limit definition of a derivative. Participants explore the calculations involved and the interpretation of the problem, debating whether it is strictly a differentiation problem or a calculation exercise.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant calculates [f(a+h)-f(a)] / h and arrives at (3h+4)/h, questioning how the book derived the answer of 3.
  • Another participant clarifies that the limit definition of the derivative leads to f'(x) = 3, but advises caution in relying solely on their solution.
  • Several participants reiterate the calculation steps, confirming that the expression simplifies to (3h+4)/h.
  • There is a suggestion that the original poster may not have learned differentiation rules yet, implying that the answer is indeed 3.
  • Some participants point out potential errors in parentheses usage that could lead to confusion in calculations.
  • One participant expresses uncertainty about whether the problem is a differentiation problem, while another asserts that it is.
  • Another participant argues that the problem may not necessarily be a differentiation problem since the limit was not explicitly taken, suggesting it could be a simple calculation exercise.
  • There is a mention that the setup resembles a typical textbook introduction to derivatives, indicating that the limit does not need to be taken to arrive at the derivative for this constant function.

Areas of Agreement / Disagreement

Participants express differing views on whether the problem is fundamentally a differentiation problem or a calculation exercise. While some assert it is a differentiation problem, others argue that the lack of an explicit limit suggests otherwise. Overall, there is no consensus on this point.

Contextual Notes

Participants note that the problem involves careful attention to parentheses and the structure of expressions, which may lead to misunderstandings in calculations. The discussion highlights the importance of clarity in mathematical notation.

CrossFit415
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f(x)=3x+2 ; find [f(a+h)-f(a)] / h

So I got (3h+4)/h

But how did the book get 3? I tried everything. It must be an error.
 
Last edited:
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Assuming that
[f(a+h)-f(a)] / h
was actually
lim(h->0) [f(a+h)-f(a)] / h
the expression above is the first principle formula to get a derivative of a function
therefore, f'(x) = 3
Please don't rely on my solution entirely and please refer to others'

by the way, I am just curious that, how did you get (3h+4)/h?
 
CrossFit415 said:
f(x)=3x+2 ; find [f(a+h)-f(a)] / h

So I got (3h+4)/h

But how did the book get 3? I tried everything. It must be an error.

How did you start?
When I calculate it, I get 3.
 
I guess you didn't learn the differentiation rules yet? Or do you have to find the answer using that formula?

But yes, the answer would be 3.
 
Well...

[3(a+h)+2-3(a)+2] / h
=(3a+3h+2-3a+2) / h
=(3h+4) / h
 
conventionally, the following expression
lim(h->0) [f(a+h)-f(a)] / h
means the slope of the function at point a.

by the way, that's a creative thinking. Nice attempt.

In that expression
a is a certain 'point' at the function f(x) and h is a certain 'distance' from the point a.
You might already know that slope of a function is defined as
delta y / delta x
so let the slope between two points, a (a, f(a)) and p (a+h, f(a+h))
then the slope is defined as
(f(a+h)-f(a)) / (a+h - a)
which is [f(a+h)-f(a)] / h
now as h approaches to 0, you will get an instaneous slope of the function at the point a
 
Last edited:
CrossFit415 said:
Well...

[3(a+h)+2-3(a)+2] / h
=(3a+3h+2-3a+2) / h
=(3h+4) / h

I see, you have left out a couple of parentheses, effectively mixing up additions and subtractions.

Try it with the following:

[(3(a+h)+2) - (3(a)+2)] / h

Cheers!
 
Yes^^
 
CrossFit415 said:
Well...

[3(a+h)+2-3(a)+2] / h
=(3a+3h+2-3a+2) / h
=(3h+4) / h

Your mistake is in the first line: f(a)=3a+2, thus -f(a)=-3a-2...
 
  • #10
I like Serena said:
I see, you have left out a couple of parentheses, effectively mixing up additions and subtractions.

Try it with the following:

[(3(a+h)+2) - (3(a)+2)] / h

Cheers!

Ahh careless mistakes again! Thank you!
 
  • #11
so it turns out to be that it is not a differntiation problem?
 
  • #12
It is a differentiation problem.
 
  • #13
Nothing tells us that it's a differentiation problem... He never took the limit of anything. For all we know, this could be an exercise in calculating with letters...
 
  • #14
I see..I was just under the impression that h approaches 0, and the derivative of that function is 3.
 
  • #15
This would typically be the set up in a math textbook just before introducing the derivative.
It starts with explaining slopes of functions over an interval h.

In this specific example the limit does not have to be taken to get the same result a derivative has, because the derivative is a constant function.
 

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