1. Partial Derivative Help

Okay so I know how to get partial derivatives. One thing that I'm having trouble with is finding f_xy, because I'm not sure if I'm actually doing it correctly.

For example, lets say:

z=xy(5-x)(y-2)

f_x=(5-2x)y(y-2)
f_y=x(5-x)(2y-2)

to my understanding.... to get f_xy you have to take the partial derivative of y and then using that function now take it into respect of x.

So the answer i got was

f_xy=(5-2x)(2y-2)

2. Hello,

Your answer is correct. However you explained the method of getting $\displaystyle f_{xy}$ in wrong way (you flipped x and y!).
In fact, $\displaystyle f_{xy} = (f_x)_y$. That is, you find $\displaystyle f_x$ then you differentiate what you get with respect to y.

Hope that helps

to my understanding.... to get f_xy you have to take the partial derivative of y and then using that function now take it into respect of x.
Well its really the other way around but i think you could find $\displaystyle f_{xy} = f_{yx}$ this time.

First of all take $\displaystyle z=xy(5-x)(y-2)$ and expand it out to get

$\displaystyle z= 5xy^2-10xy-x^2y^2+2x^2y$

which is really

$\displaystyle f(x,y,z) = 5xy^2-10xy-x^2y^2+2x^2y-z$

Now taking the derivative with respect to $\displaystyle x$ you get

$\displaystyle f_{x} = 5y^2-10y-2xy^2+4xy$

Now take this $\displaystyle f_{x}$ with respect to $\displaystyle y$

$\displaystyle f_{xy} = 10y-10-4xy+4x$

And you are done.