Partial derivatives with respect to other variables?

[allhalf425]

Okay, I'm familiar with taking the partial derivative of an equation in respect to a certain variable (ie. partial derivative with respect to x, y, or z), but what does it mean when it says something like this:

Given F(x, y, z) = 5x[sup:k57cp2b3]2[/sup:k57cp2b3] - 6y[sup:k57cp2b3]2[/sup:k57cp2b3] + 3z[sup:k57cp2b3]2[/sup:k57cp2b3] - 4xy - 2yz + 8xz = 0, with z = f(x, y), determine the "partial derivative of z with respect to x" and "the partial derivative of z with respect to y" as functions of x, y, and z.


Any help would be appreciated. Thanks.

 

[BigGlenntheHeavy]

\(\displaystyle F(x,y,z) \ = \ f(x,y)-z, \ ergo \ F_x(x,y,z) \ = \ f_x(x,y) \ and \ F_y(x,y,z) \ = \ f_y(x,y)\)

 

[allhalf425]

Okay, so if I have the partial derivatives of the original function in respect to x and y, I have the partial derivatives of the new function f(x, y)? How do I then obtain the final equation f(x, y)? Integration?