I need some clarification on how to get the and on:

I know that the solution is and but I have no idea how they got to it.

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- Mar 29th 2008, 12:56 PMlllllpartial derivative question
I need some clarification on how to get the and on:

I know that the solution is and but I have no idea how they got to it. - Mar 29th 2008, 02:18 PMgalactus
Let's do

We can use implicit differentiation letting y be a constant.

- Mar 29th 2008, 02:25 PMmr fantastic
Use

*implicit differentiation*(z van be treated as an implicit function of x and y):

Differentiate both sides partially with resepct to x:

.

The chain rule has been used on the left hand side to differentiate z^2 and the product rule has been used on the right to differentiate the product.

Now re-arrange the result to make the subject.

Same idea when getting the partial derivative with respect to y.

Note that you can re-arrange the given rule to make z the subject and then take the partial derivative with respect to x in the usual way.