Year 12s: what will be the first way you research your future options? >>

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Can anyone answer this?

http://www.thestudentroom.co.uk/showthread.php?t=1955808&p=36954684#post36954684

Well, as you suggested, it's not very clear what f is supposed to be. My best guess would be that f is supposed to be z (because it was defined as a function of x and y) and that they're asking for implicit differentiation. You could also isolate z and then use a quotient rule, which would get you to the same place.

xyz = x - y + z

For example, if you take the partial in x you get

yz + xy\frac{\partial z}{\partial x} = 1 + \frac{\partial z}{\partial x}

\frac{\partial z}{\partial x} = \frac{yz - 1}{1 - xy}

If you want to get rid of the z, you can write

z = \frac{x - y}{xy - 1}

and then substitute back in to get

\frac{\partial z}{\partial x} = \frac{y^2 - 1}{(xy - 1)^2}

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