Calculus III - Chain Rule

Section 13.6 : Chain Rule

8. Determine formulas for $\displaystyle \frac{{\partial w}}{{\partial t}}$ and $\displaystyle \frac{{\partial w}}{{\partial u}}$ for the following situation.

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

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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 $x$, $y$ and $z$. From $x$ there is a branch to $t$; from $y$ there are branches to $u$, $v$ and $p$; and from $z$ there are branches to $v$ and $p$. Each $v$ then branches on to $r$ and $u$ and each $p$ branches on to $t$ and $u$. The last row of derivatives uses subscript notation to keep the labels readable and the variables we are differentiating with respect to are colored.$

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.

Also, because the last “row” of branches was getting a little close together we switched to the subscript derivative notation to make it easier to see which derivative was associated with each branch.

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

\[\frac{{\partial w}}{{\partial t}} = \frac{{\partial w}}{{\partial x}}\frac{{\partial x}}{{\partial t}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial p}}\frac{{\partial p}}{{\partial t}} + \frac{{\partial w}}{{\partial z}}\frac{{\partial z}}{{\partial p}}\frac{{\partial p}}{{\partial t}}\] \[\frac{{\partial w}}{{\partial u}} = \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial u}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial v}}\frac{{\partial v}}{{\partial u}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial p}}\frac{{\partial p}}{{\partial u}} + \frac{{\partial w}}{{\partial z}}\frac{{\partial z}}{{\partial v}}\frac{{\partial v}}{{\partial u}} + \frac{{\partial w}}{{\partial z}}\frac{{\partial z}}{{\partial p}}\frac{{\partial p}}{{\partial u}}\]