Basic bivariate question

  • #1
given f(x,y) = (xy+(x^3))/(x^2+xy)

I want to calculate df/dx (i.e. first partial derivative wrt x) at the origin.

f is undefined there, but i can still do it, right?

so, we get (x^2 + 2xy -y)/(x^2 +2xy +y^2)

I'm not really sure of where to go with this, though.

I think the basic definition of the derivative, i.e. lim (h->0) ((x+h)y + (x+h)^3)/((x+h)^2 + (x+h)y) could be the way to go.

Any advice would be appreciated.

The only specific techniques i understand are the "sandwich rule", Taylor polynomials, as well as basic differentiation etc, and the 1-var stuff like l'Hopital's rule. So please don't give me something like the epsilon-delta stuff for limits or anything else i won't understand.
 

Answers and Replies

  • #2
HallsofIvy
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1. IF a function is not defined at a point then it does not have partial derivatives there! When yuou talk about "lim (h->0) ((x+h)y + (x+h)^3)/((x+h)^2 + (x+h)y) " you seem to be assuming that f(0,0)= 0. Is that given? If so, then f certainly is defined there!

2. Note that you want to find the derivative AT (0,0). The partial derivative (with respect to x) is calculated holding y constant so you are really concerned with the derivative of f(x,0)= x3/x2 which is equal to x as long as x is not 0- and, assuming that f(0,0)= 0, is also equal to 0 at x= 0 and so is equal to x for all 1. Surely you can find the derivative of that!
 

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