Calculus III - Chain Rule

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

$A tree diagram for the chain rule. At the top is $w$, which branches down to $x$ and $y$. From $x$ there are branches to $p$, $q$ and $s$, with $p$ continuing on to $t$ and $s$ branching on to $u$ and $v$. From $y$ there are branches to $v$, $u$ and $p$, with $p$ again continuing on to $t$. The variables we are differentiating with respect to are colored to make the branches easier to follow.$

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.

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

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