What is ?

I've tried experimenting but I don't know how to simplify it!! Can someone please tell me where to go?

THANKS :)

Printable View

- Mar 13th 2010, 01:53 AMDivideBy0Partial derivative of a function
What is ?

I've tried experimenting but I don't know how to simplify it!! Can someone please tell me where to go?

THANKS :) - Mar 13th 2010, 02:11 AMMiss
- Mar 13th 2010, 02:31 AMHallsofIvy
- Mar 13th 2010, 02:41 AMDivideBy0
Hold on, are you saying that

?

This is what I also got when I tried it, but it can't be right, can it? - Mar 13th 2010, 02:45 AMMiss
- Mar 13th 2010, 03:07 AMtom@ballooncalculus
Just in case a picture helps...

http://www.ballooncalculus.org/asy/diffChain/xy.png

... where

http://www.ballooncalculus.org/asy/chain.png

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x (in the single variable case we would take this as read but here we label the line just to be clear), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Hope this helps, or doesn't further confuse...

_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote! - Mar 13th 2010, 03:26 AMDivideBy0
Thanks for the diagrams Tom, I think I understand how the chain rule works

However my main problem is that if

, then

that is a recursive relationship!

So

So ?

I don't see why this should follow. And indeed I don't want it to have to follow.

Moreover, say you have , then

Clearly unless , which I don't want to assume.

There must be a solution that doesn't have is there?

Thanks again - Mar 13th 2010, 03:35 AMtom@ballooncalculus
Sorry I didn't notice you'd written

before - which should be

You could mark the distinction with a u-sub (for xy) or notice that the diagram says we differentiate down the dashed line w.r.t. the dashed balloon expression.

e.g... (good example)

http://www.ballooncalculus.org/asy/diffChain/xy1.png

Edit:

We could also label the dashed line to show what it's w.r.t...

http://www.ballooncalculus.org/asy/diffChain/xy2.png

... although this is always the case. - Mar 13th 2010, 04:09 AMDivideBy0
thanks again tom,

I think I understand it now :)