Homework Help: Why is this true?

pivoxa15 · Mar 17, 2007

1. The problem statement, all variables and given/known data
(d/dx)c=(d/da)c(da/dx)

where c=f(a(x))

3. The attempt at a solution
It seems correct because the da cancels but how do you get this from first principles?

radou · Mar 17, 2007

You may want to investigate the chain rule.

pivoxa15 · Mar 17, 2007

Could you possibly be a little bit more specific?

neutrino · Mar 17, 2007

Chain Rule: http://mathworld.wolfram.com/ChainRule.html

HallsofIvy · Mar 17, 2007

pivoxa15 said: ↑

1. The problem statement, all variables and given/known data
(d/dx)c=(d/da)c(da/dx)

where c=f(a(x))

3. The attempt at a solution
It seems correct because the da cancels but how do you get this from first principles?

"da/dx" and "dc/da" are NOT fractions so it is not correct to say that the "da" cancels!

radou said: ↑

You may want to investigate the chain rule.

pivoxa15 said: ↑

Could you possibly be a little bit more specific?

Are you saying you have never heard of the chain rule? You are being asked to prove the chain rule!
[tex]\frac{dc}{dx}= \lim_{h\rightarrow 0} \frac{c(x+h)- c(x)}{h}[/tex]
"Before" the limit, this is a fraction- you can cancel before the limit but be careful, this is not a trivial proof.

pivoxa15 · Mar 17, 2007

I have heard of the chain rule.

In fact I realise that the chain rule is just what I stated (d/dx)c=(d/da)c(da/dx)

pivoxa15 · Mar 18, 2007

HallsofIvy said: ↑

"da/dx" and "dc/da" are NOT fractions so it is not corret to say that the "da" cancels!

But physicists and applied mathematicians like to treat them as fractions in the limit. Is it okay to treat them as fractions and specify "in the limit"?

HallsofIvy · Mar 19, 2007

pivoxa15 said: ↑

But physicists and applied mathematicians like to treat them as fractions in the limit. Is it okay to treat them as fractions and specify "in the limit"?

Yes, because the derivative is the limit of a fraction, you can always treat them "like" a fraction- that's one of the advantages of the dy/dt notation over f '. And, in fact, it is motivation for defining the "differentials", dx and dy= f '(x) dx.

My point here was that, if you are being asked to prove the chain rule, you can't just "treat dy/dx like a fraction" since the chain rule is part of what allows us to do that.

Gib Z · Mar 19, 2007

If i was a pedantic bastard, I would say perhaps pivoxa15 was speaking of using hyperreals? Thank god im not :D

HallsofIvy · Mar 19, 2007

If you do, I'll beat you around the head and shoulders with a 2 by 4!