derivative w.r.t. a function


by ledol83
Tags: derivative, function
ledol83
ledol83 is offline
#1
Aug2-07, 10:47 PM
P: 12
Hi, i have a question on taking derivative w.r.t. to a function (instead of an independent variable). Actually i saw an excellent post on this same forum but that one was about a single variable.

My question is: f is function of x and y, and z is some other function also dependent on x and y, so is the following correct?

df/dz=df/dx*dx/dz+df/dy*dy/dz

it differs from the classic chain rule in the sense that z is actually a function (not an independent var), so i am not sure about this.

I appreciate so much for any comment!
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
lalbatros
lalbatros is offline
#2
Aug3-07, 01:49 AM
P: 1,235
First, go back to how the derivative wrt to a function is defined. (functional derivative)
Second, be more precise about your specific question.
As I understood, f is a function of x and y: f(x,y)
therefore, if you confirm that, I would say the df/dz = 0 .
ledol83
ledol83 is offline
#3
Aug3-07, 09:16 AM
P: 12
Hi actually i have f(x,y) and z(x,y) and was just wondering if this is true:

df/dz=df/dx*dx/dz+df/dy*dy/dz

thanks a lot!!

lalbatros
lalbatros is offline
#4
Aug3-07, 03:09 PM
P: 1,235

derivative w.r.t. a function


No this cannot be true, since this has no meaning.
Tell us what you think the meaning of df/dz would be, maybe then we can help.
ledol83
ledol83 is offline
#5
Aug3-07, 03:21 PM
P: 12
i have realized that what i posed was not meaningful. i am now thinking over my problem again.. thanks!
ice109
ice109 is offline
#6
Aug3-07, 03:23 PM
P: 1,705
how could you have a derivative of a function with respect to another function that the first function is not a function of
HallsofIvy
HallsofIvy is offline
#7
Aug3-07, 04:22 PM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,898
If f(x) is a function of x and g(x) is a function of x, you can surely write f as a function of g.

In particular,
[tex]\frac{df}{dg}= \frac{df}{dx}\frac{dx}{dg}= \frac{\frac{df}{dx}}{\frac{dg}{dx}}[/tex]
ice109
ice109 is offline
#8
Aug3-07, 05:34 PM
P: 1,705
Quote Quote by HallsofIvy View Post
If f(x) is a function of x and g(x) is a function of x, you can surely write f as a function of g.

In particular,
[tex]\frac{df}{dg}= \frac{df}{dx}\frac{dx}{dg}= \frac{\frac{df}{dx}}{\frac{dg}{dx}}[/tex]
not that i'm doubting you personally but i don't see where this comes from; proof?
lalbatros
lalbatros is offline
#9
Aug5-07, 02:55 PM
P: 1,235
"not that i'm doubting you personally but i don't see where this comes from; proof?"

The first point is to know when we are talking about f1=f(g) or f2=f(x),
The second point is about the class of functions considered,
Otherwise, this is trivial (assuming dg is smooth):

f'(g) = f(g+dg)/dg = f(g(x)+dg(x))/dg(x) = f(g(x) + g'(x) dx) / (g'(x) dx) = f'(x)/g'(x)
AiRAVATA
AiRAVATA is offline
#10
Aug5-07, 07:55 PM
P: 173
Quote Quote by ice109 View Post
not that i'm doubting you personally but i don't see where this comes from; proof?
It is a consequence of the chain rule and the inverse function theorem. Maybe you should do some reading too instead of making fun of peoples questions. Better try to help or keep out.


Register to reply

Related Discussions
Need Help With Derivative Function Calculus & Beyond Homework 6
Lie derivative of a function Differential Geometry 5
derivative of function Calculus & Beyond Homework 5
Derivative of a function with ln Calculus & Beyond Homework 7
derivative of the function Calculus 4