# Derivative question

## Main Question or Discussion Point

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?

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) thats why im 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.

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 doesnt 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:
You will always get the same answer, no matter which method you use.

CompuChip
Homework Helper
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:
$$x^2 = x^3 / x$$
so if $f(x) = x^3, f'(x) = 3 x^2$ and $g(x) = x, g'(x) = 1$ so
$$f'(x) / g'(x) = 3 x^2 / 1 = 3 x^2 \stackrel{!}{\neq} 2 x.$$

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.

"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