# Derivative question

1. Mar 30, 2008

### alpha01

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?

2. Mar 30, 2008

### montoyas7940

Are you using the quotient rule?

3. Mar 30, 2008

### grmnsplx

montoyas7940 said it: quotient rule.

Unless you are specifically asked to use the definition of the derivative, but I can't imagine why?

4. Mar 31, 2008

### alpha01

It specifically asks to use the definition (not the quotient rule) thats why im a bit confused.

5. Mar 31, 2008

### Sourabh N

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.

6. Mar 31, 2008

### alpha01

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?

7. Mar 31, 2008

### Sourabh N

absolutely no... you can't do that.

8. Mar 31, 2008

### alpha01

thats what i initially did, but i didnt think it was right. thanks, ill do this again.

9. Mar 31, 2008

### alpha01

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: Mar 31, 2008
10. Mar 31, 2008

### Sourabh N

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

11. Mar 31, 2008

### CompuChip

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.

12. Mar 31, 2008

### meaw

"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