Paul's Online Notes
Home / Calculus III / Partial Derivatives / Chain Rule
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 13.6 : Chain Rule

1. Given the following information use the Chain Rule to determine $$\displaystyle \frac{{dz}}{{dt}}$$ .

$z = \cos \left( {y\,{x^2}} \right)\,\hspace{0.5in}x = {t^4} - 2t,\,\,\,\,y = 1 - {t^6}$ Show Solution

Okay, we can just use the “formula” from the notes to determine this derivative. Here is the work for this problem.

\begin{align*}\frac{{dz}}{{dt}} & = \frac{{\partial z}}{{\partial x}}\frac{{dx}}{{dt}} + \frac{{\partial z}}{{\partial y}}\frac{{dy}}{{dt}}\\ & = \left[ { - 2xy\sin \left( {y{x^2}} \right)} \right]\left[ {4{t^3} - 2} \right] + \left[ { - {x^2}\sin \left( {y{x^2}} \right)} \right]\left[ { - 6{t^5}} \right]\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 2\left( {{t^4} - 2t} \right)\left( {1 - {t^6}} \right)\left( {4{t^3} - 2} \right)\sin \left( {\left( {1 - {t^6}} \right){{\left( {{t^4} - 2t} \right)}^2}} \right) + 6{t^5}{{\left( {{t^4} - 2t} \right)}^2}\sin \left( {\left( {1 - {t^6}} \right){{\left( {{t^4} - 2t} \right)}^2}} \right)}}\end{align*}

In the second step we added brackets just to make it clear which term came from which derivative in the “formula”.

Also, we plugged in for $$x$$ and $$y$$ in the third step just to get an equation in $$t$$. For some of these, due to the mess of the final formula, it might have been easier to just leave the $$x$$’s and $$y$$’s alone and acknowledge their definition in terms of $$t$$ to keep the answer a little “nicer”. You should probably ask your instructor for his/her preference in this regard.