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MathGroup Archive 2007

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Re: Is it possible?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72965] Re: Is it possible?
  • From: dh <dh at metrohm.ch>
  • Date: Fri, 26 Jan 2007 07:46:43 -0500 (EST)
  • References: <epa2ic$iqn$1@smc.vnet.net>

Hi,
First, note that it does not matter how v depends on a internally. You 
have a scalar function of one argument: y=f[x]. All you are asking for 
is the derivative of the inverse function of a given function . The key 
word here is "inverse function". The derivative of the inverse function 
is the inverse of the derivative. Consider dy=f'[x] dx and you want 
dx/dy=(dy/dx)^-1= 1/f'[x], This looks simple but there is a catch. The 
formula gives you dx/dy as a function of x. If this is fine your are 
done. However if you need a function of y you will need the inverse 
function to f. In general this is hard, but you may be lucky and have a 
simple case, all depends on the actual function. For the general case, 
you can search for the key word "lagrage inversion theorem".
Further, obviously, inversion is impossible in a region where f' is zero 
somewhere.

Daniel

Travelmate wrote:
> Hi,
> 
> I'd really appreciate your help...
> 
> I'been asked to perform a kind of analysis I've never done before..and 
> I'm not even sure that it's possible.
> 
> Suppose you have value function V=v[x(a,b,c,d),y(....),z(......)]
> 
> The key-parameter is 'a' and it turns out that the first and second 
> (partial) derivatives of V with respect to 'a' are negative and positive 
> respectively. (V is continous)
> 
> Now there's a problem: I've been asked to invert V(a) (keeping b,c and d 
> constant)and to study the derivatives of this inverse function with 
> respect to b, c and d. I'm not sure it's possible to do such a thing. 
> ANd even if it is possible, I do not know how to start.
> 
> Could you be so kind to tell me:
> 1) if it is actually possible to perform this study;
> 2) if the answer is yes, could you suggest me a starting point or any 
> reference?
> 
> Thank you in advance
> 


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