# Partial Derivatives

1. Jan 20, 2009

### KillerZ

1. The problem statement, all variables and given/known data

Use the definition of partial deriviatives as limits to find fx(x,y) and fy(x,y).

2. Relevant equations

f(x,y) = $$\frac{x}{x + y^{2}}$$

3. The attempt at a solution

I don't think this is right because I think I should have an answer of 1.

fx(x,y) = lim h-> 0 [f(x+h,y) - f(x,y)]/h

=lim h->0 [(x+h)/(x+h+y^2) - x/(x+y^2)]/h
=lim h->0 [(x+h)/(x+h+y^2) - x/(x+y^2)]*1/h
=lim h->0 (x+h)/(xh+h^2+(y^2)h) - x/(xh+(y^2)h)
=lim h->0 ((x/h)+1)/(x+h+y^2) - (x/h)/(x+y^2)
=1/(x+y^2)

2. Jan 20, 2009

### Dick

I don't think the answer is 1. Look at your last line. You got things like (x/h). If you take lim h->0, that goes to infinity. You need to do enough algebra to cancel the h in the denominator before you can find a sensible limit. Combine (x+h)/(x+h+y^2) - x/(x+y^2) into single fraction and simplify the numerator before you take the limit.

3. Jan 20, 2009

### Staff: Mentor

fx(x, y) is not equal to 1.

I think you have an error in this line:
=lim h->0 ((x/h)+1)/(x+h+y^2) - (x/h)/(x+y^2)

I don't understand how you got to this expression from the one just before it.

After taking the limit, you should end up with y^2/(x + y^2)^2