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July 2nd, 2017, 04:29 PM   #1
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implicit partial differentiation

Hello!

This is an example problem in my textbook.



I am stuck on understanding the second line. Everything makes absolutely perfect sense until they went ahead and tacked on that very last term the $+\,6xy\dfrac{\partial z}{\partial x}$. Perhaps the product rule was used? I am not sure. I would really appreciate some clarification on that line if it's possible!

Thank you!

Jacob

Last edited by skipjack; July 2nd, 2017 at 06:23 PM.
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July 2nd, 2017, 06:01 PM   #2
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Hello Jacob, the result is as follows. The key is to understand that $y$ is a constant here. From the first line, call $\alpha=\partial z/\partial x$, then we have

\begin{eqnarray}
3x^{2} + 3z^{2}\alpha + 6yz +6yx\alpha=0 \to \alpha(6xy + 3z^{2})=-3x^{2}-6yz
\end{eqnarray}
then dividing we end up with
\begin{eqnarray}
\alpha=\frac{\partial z}{\partial x}=-\frac{3x^{2}+6yz}{6xy + 3z^{2}}=-\frac{x^{2}+2yz}{z^{2}+2xy}
\end{eqnarray}
Hope it is clear now, same procedure is done for $\partial z/\partial y$ but keeping $x$ as a constant.

Last edited by skipjack; July 2nd, 2017 at 06:27 PM.
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July 2nd, 2017, 06:04 PM   #3
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Quote:
Originally Posted by SenatorArmstrong View Post
Hello!

This is an example problem in my textbook.

http://i.imgur.com/zqCtZcZ.jpg

I am stuck on understanding the second line. Everything makes absolutely perfect sense until they went ahead and tacked on that very last term the $+\,6xy\dfrac{\partial z}{\partial x}$. Perhaps the product rule was used? I am not sure. I would really appreciate some clarification on that line if it's possible!

Thank you!

Jacob
Ok I saw now what you mean, precisely is the product rule, so that $\partial_{x}(6xyz)= 6yz + 6xy(\partial _{x}z)$
Thanks from SenatorArmstrong

Last edited by skipjack; July 2nd, 2017 at 06:24 PM.
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July 3rd, 2017, 04:01 PM   #4
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Quote:
Originally Posted by nietzsche View Post
Ok I saw now what you mean, precisely is the product rule, so that $\partial_{x}(6xyz)= 6yz + 6xy(\partial _{x}z)$
Thank you for walking me through this. Makes crystal clear sense now!
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