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Section 13.6 : Chain Rule

7. Determine formulas for \(\displaystyle \frac{{\partial w}}{{\partial t}}\) and \(\displaystyle \frac{{\partial w}}{{\partial v}}\) for the following situation.

\[w = w\left( {x,y} \right)\hspace{0.5in}x = x\left( {p,q,s} \right),\,\,\,\,y = y\left( {p,u,v} \right),\,\,\,\,s = s\left( {u,v} \right),\,\,\,\,p = p\left( t \right)\]

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Start Solution

To determine the formula for these derivatives we’ll need the following tree diagram.

Some of these tree diagrams can get quite messy. We’ve colored the variables we’re interested in to try and make the branches we need to follow for each derivative a little clearer.

Show Step 2

Here are the formulas we’re being asked to find.

\[\frac{{\partial w}}{{\partial t}} = \frac{{\partial w}}{{\partial x}}\frac{{\partial x}}{{\partial p}}\frac{{dp}}{{dt}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial p}}\frac{{dp}}{{dt}}\hspace{0.5in}\frac{{\partial w}}{{\partial v}} = \frac{{\partial w}}{{\partial x}}\frac{{\partial x}}{{\partial s}}\frac{{\partial s}}{{\partial v}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial v}}\]