Why is this true?

You may want to investigate the chain rule.

 

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

 

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

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.

 

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.

 

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

 

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