Find Derivative of f(x) = x+2 / x-2

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Discussion Overview

The discussion revolves around finding the derivative of the function f(x) = (x + 2) / (x - 2). Participants explore different methods for calculating the derivative, including the quotient rule and the definition of the derivative, while addressing the complexity involved in each approach.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant expresses confusion about whether to simplify the function before finding its derivative.
  • Another participant suggests using the quotient rule for differentiation.
  • A participant notes that the question specifically asks for the definition of the derivative, which adds to the confusion.
  • Some participants argue that using the definition of the derivative is straightforward, involving substitution and simplification.
  • One participant proposes breaking the function into its components and finding the derivatives of each separately, but this is challenged by another participant.
  • A participant questions whether the same methods can be applied to a different function, f(x) = (x^2 - 1) / x, and whether the results would be consistent across methods.
  • Another participant asserts that all methods should yield the same result, emphasizing that the choice of method depends on convenience.
  • One participant attempts to clarify the correct formula for differentiation in this context, referencing the product and quotient rules.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to use for finding the derivative. There are competing views on whether to apply the quotient rule or the definition of the derivative, and some participants express differing opinions on the complexity of each approach.

Contextual Notes

Some participants mention the potential for confusion when the problem does not specify which method to use, indicating that the choice may depend on individual preferences or the specific context of the question.

alpha01
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Im trying to find the derivative of f(x) = x+2 / x-2

I know the formula to apply to this but it get quite messy because this example is a fraction.

Maybe i need to put function f(x) in a more simplier form before attempting to find its derivative?
 
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Are you using the quotient rule?
 
montoyas7940 said it: quotient rule.

Unless you are specifically asked to use the definition of the derivative, but I can't imagine why?
 
It specifically asks to use the definition (not the quotient rule) that's why I am a bit confused.
 
I didn't find any serious complexity to do it using the definition. Just plugin the values f(x+h) and f(x), some cross multiplication, some cancellation and you are done.
 
maybe break it up into f (x) = x +2 and g (x) = x - 2, find the derivatives of f(x) and g(x) using the definition, then solve f '(x) / g '(x)?

is that what the question means?
 
absolutely no... you can't do that.
 
Sourabh N said:
I didn't find any serious complexity to do it using the definition. Just plugin the values f(x+h) and f(x), some cross multiplication, some cancellation and you are done.

thats what i initially did, but i didnt think it was right. thanks, ill do this again.
 
can i also do this for f(x) = (x^2 -1) / x ?

(for this question it doesn't say which rule to use)

or should i just use the quotient rule for this?

i mean, i should get the same answer if i use the quotient rule or the definition of a derivative for this one?
 
Last edited:
  • #10
You will always get the same answer, no matter which method you use. :wink:
 
  • #11
alpha01 said:
maybe break it up into f (x) = x +2 and g (x) = x - 2, find the derivatives of f(x) and g(x) using the definition, then solve f '(x) / g '(x)?

is that what the question means?

Finding the derivative of x^2 with this method would then work like:
[tex]x^2 = x^3 / x[/tex]
so if [itex]f(x) = x^3, f'(x) = 3 x^2[/itex] and [itex]g(x) = x, g'(x) = 1[/itex] so
[tex]f'(x) / g'(x) = 3 x^2 / 1 = 3 x^2 \stackrel{!}{\neq} 2 x.[/tex]

When they don't say which rule to use you just use the one which is the most convenient. It can be rigorously proven that they all give the same answer (of course, they should, otherwise you wouldn't be allowed to use them in the first place). And the definition is never the most convenient one, if you know the sum, product, quotient and chain rules.
 
  • #12
"maybe break it up into f (x) = x +2 and g (x) = x - 2, find the derivatives of f(x) and g(x) using the definition, then solve f '(x) / g '(x)? "

the right formula in this case will be

f'(x).g(x) - f(x).g'(x) / g'(x) ^ 2.

for the example with x^2 = X^3 / X , it will be
3x^2. x - 1. x^3 / x^2 = 2x^3 / x^2 = 2x
 

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