Quotient Rule (In what order do you choose which to differentiate)

[Jani08]

Homework Statement

The Derivative of x-2/(x^2-1)

Can you show me how to get the derivative using the quotient rule and tell me if the order matters? I always thought that the order never mattered until I got a wrong graph because it seems I got the wrong derivative.

Homework Equations

The Attempt at a Solution

So which one is correct?
If I plug in 0.1 on each of these one gives me a + answer and the other -

I got -x^2+4x-1/(x^2-1)^2
and
x^2-4x+1/(x^2-1)^2

At first I thought these where the same but then as I was plugging in random point to see if it increased or decreased at the critical points I did not get the same results. It seems that I've used the quotient rule correctly but the answer changes depending on which you choose first to differentiate.

 

[Hurkyl]

You can evaluate a compound expression in any order you want, as long as you do it correctly. What do you mean by "order matters"? I have a suspicion you are comparing two different formulas, rather than comparing two different ways to evaluate a single formula.

 

[Jani08]

Well you know how we have (f/g)

what is (f/g)' ?

is it f'g-g'f/g^2

or g'f-f'g/g^2

OR does it even matter? I thought it didn't matter until I came to this problem

 

[gb7nash]

Well you know how we have (f/g)

what is (f/g)' ?

is it f'g-g'f/g^2

or g'f-f'g/g^2

OR does it even matter? I thought it didn't matter until I came to this problem

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

 

[Hurkyl]

Well you know how we have (f/g)

what is (f/g)' ?

is it f'g-g'f/g^2

or g'f-f'g/g^2

OR does it even matter? I thought it didn't matter until I came to this problem

Well, the two formulas

$$\frac{f'g - g'f}{g^2}$$​

and

$$\frac{g'f - f'g}{g^2}$$​

are formally different. Of course, that doesn't mean they are actually unequal -- have you thought about trying to prove/disprove their equality? It would be one of those "prove or disprove the following identity" kind of problems. I guess you have already done the "disproof", since you found a counterexample. Do you remember why you thought it didn't matter?


Incidentally, what you wrote is technically wrong: the meaning of

$$f'g - g'f/g^2$$​

is

$$f'g - \frac{g'f}{g^2}$$​

and not

$$\frac{f'g - g'f}{g^2}$$​

Even if you know what you mean, you really should add parentheses when talking to other people:

$$(f'g - g'f)/g^2$$​

because they might not realize that you are abusing notation. Actually, I would advise using parentheses even when you're writing just for yourself -- I've seen many people make mistakes because they got confused on how things were grouped.

 

[Mark44]

Note the parentheses that gb7nash added. When you write x-2/(x^2-1), people would interpret this as x - (2/(x^2 - 1)), which is probably not what you intended.

Also, (f'g-g'f)/g^2 and (g'f - f'g)/g^2 have opposite signs.

 

[vela]

I suspect you were confused because the order doesn't matter with the product rule. That's because in the product rule, you're adding the terms, and addition is commutative. If you differentiate f first, you get f'g+g'f, and if you differentiate g first, you get g'f+f'g. Since the order doesn't matter for addition, those two expressions are equal.

With the quotient rule, however, there's a subtraction rather than an addition. As you know, the order does matter for subtraction: a-b isn't the same as b-a. So for the quotient rule, you need to remember that you differentiate f first and g second.

If you can't remember that, just write f/g as f(g^-1) and use the product and chain rules to rederive the quotient rule.

 

[Jani08]

Hurkyl thanks for tips, I guess I should type it the same way I type it in my graphing calculator.

Vela you got it spot on, now that I look at it the way of properties it does make sense. Hah I finally got it...after a whole year doing this thing..I was always just "hoping" I had the right order never really tackled it.

Thanks everyone!