Thread: 2nd Order Partial Derivitive

1. 2nd Order Partial Derivitive

Hello.

I have a real issue with fractions, it seems! can anyone help me understand how to find the 2nd order derivitives for
z=xy/(x+y)

thanks for any help

2. There are four to find. $\displaystyle \frac{\partial ^2z}{\partial x^2}$, $\displaystyle \frac{\partial ^2z}{\partial y^2}$, $\displaystyle \frac{\partial ^2z}{\partial xy}$ and $\displaystyle \frac{\partial ^2z}{\partial yx}$

You need to find $\displaystyle \frac{\partial z}{\partial x}$ and $\displaystyle \frac{\partial z}{\partial y}$ first, use the quotient rule.

3. Yeah sorry I know, and I know how to do partial derivitives, its just that I always get my numbers messed up when I have fractions, I'll try again, but for all I know I'm not even using the quotient rule correctly!

4. okay, so (xy)/(x+y).
Use quotient rule
(y).(x+y)-(xy).(y) / (x+y)^2... is that right??? now what?

5. for $\displaystyle y = \frac{u}{v}$ then $\displaystyle y' = \frac{vu'-uv'}{v^2}$

I get

$\displaystyle \frac{\partial z}{\partial x} = \frac{(x+y)(xy)'-(xy)(x+y)'}{(x+y)^2}= \frac{(x+y)y-(xy)\times 1}{(x+y)^2}= \frac{y^2}{(x+y)^2}$