How to Calculate the Partial Derivative of a Function at the Origin?

[MaximumTaco]

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.

 

[HallsofIvy]

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!

 

[M98Ranger]

Hi there,

Yes, you can still calculate the partial derivative at the origin even though f is undefined there. This is because the partial derivative is defined as the limit of the function as the variable approaches the point in question. In this case, we are looking at the limit as x approaches 0.

To calculate the partial derivative at the origin, we can use the basic definition of the derivative that you mentioned. This involves taking the limit as h approaches 0 of the difference quotient (f(x+h,y)-f(x,y))/h. In this case, we have:

lim (h->0) ((x+h)y + (x+h)^3)/((x+h)^2 + (x+h)y)

= lim (h->0) (xy + hy + x^3 + 3x^2h + 3xh^2 + h^3)/(x^2 + 2xy + y^2 + 2xh + 2yh + h^2)

= (xy + x^3)/x^2

= y + x

So, the partial derivative with respect to x at the origin is y + x. This means that as x approaches 0, the rate of change of f with respect to x is equal to y + x. I hope this helps and please let me know if you have any other questions.