Messy partial differentials with chain rule.

[phewy13]

Homework Statement

the problem asks: Find \deltaf/\deltax and \deltaf/\deltay at x=1 and y=2 if z=f(x,y) is defined implicitly by 2x^{}2y/z + 3z/xy - xy\sqrt{}z = 3. Note that (1,2,4) is a point on the surface.

Homework Equations

Im not really sure how to approach this one.

The Attempt at a Solution

i started off by saying that \deltaf/\deltax is equal to \deltaz/\deltax and the same thing for y. i went through and found the partial derivatives of the above equation and it turned out really messy, any help would be greatly appreciated.

 

[hotcommodity]

You're right, \frac{\partial f}{\partial x} is the same as \frac{\partial z}{\partial x}.

First we want to find the partial derivative with respect to x. Remember in calculus 1 when we differentiated implicitly, we would place a y', or a dy/dx, or something to denote the derivative of y, everytime we had to take the derivative of a term containing y. We pretty much do the same thing when we take the partial derivative of a function of two or more variables. Everytime you take the derivative of a term with z in it, simply take the derivative and place a \frac{\partial z}{\partial x} next to it.

For example, if I have:

x^{2}z + y + z^2 = 10

I differentiate both sides with respect to x:

x^{2}\frac{\partial z}{\partial x} + 2xz + \frac{\partial z}{\partial x}2z = 0

Get all the \frac{\partial z}{\partial x} terms on one side:

\frac{\partial z}{\partial x} = \frac{-2xz}{x^{2} + 2z}

And now you can plug in values. Does this help?