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 March 15th, 2009, 06:57 PM #1 Newbie   Joined: Mar 2009 Posts: 5 Thanks: 0 Multi variable chain rule Given q = f(r,s) r(t) = 3 + 2t s(t) = 2 - 6t What is q '(0) Any help would be nice.
 March 16th, 2009, 01:15 PM #2 Senior Member   Joined: Dec 2008 Posts: 251 Thanks: 0 Re: Multi variable chain rule The Multivariable Chain Rule states: $\frac{df}{dt}\,=\,\frac{\partial f}{\partial r}\frac{dr}{dt}\,+\,\frac{\partial f}{\partial s}{\frac{ds}{dt}}.$ This formula holds because the value of $f$ is impacted by the changes in both $r$ and $s$ as we increase $t$. In our case, $\begin{eqnarray*} \frac{dr}{dt} &=& \frac{d}{dt}(3\,+\,2t) &=& \frac{d}{dt}(3)\,+\,\frac{d}{dt}(2t) &=& 0\,+\,2 &=& 2 \\ \frac{ds}{dt} &=& \frac{d}{dt}(2\,-\,6t) &=& \frac{d}{dt}(2)\,-\,\frac{d}{dt}(6t) &=& 0\,-\, 6 &=& -6. \end{eqnarray*}$
 March 16th, 2009, 07:50 PM #3 Newbie   Joined: Mar 2009 Posts: 5 Thanks: 0 Re: Multi variable chain rule So with information given we cannot find q '(0) because we don't know the partial derivatives?
 March 17th, 2009, 05:14 AM #4 Senior Member   Joined: Dec 2008 Posts: 251 Thanks: 0 Re: Multi variable chain rule They might want you to include $\frac{\partial q}{\partial r}(0)$ and $\frac{\partial q}{\partial s}(0)$ in the formula.
 March 17th, 2009, 10:24 AM #5 Newbie   Joined: Mar 2009 Posts: 5 Thanks: 0 Re: Multi variable chain rule Alright thanks for the help.

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