Skip to main content
\(\newcommand{\Z}{\mathbb{Z}} \newcommand{\reals}{\mathbb{R}} \newcommand{\real}[1]{\mathbb{R}^{#1}} \newcommand{\fe}[2]{#1\mathopen{}\left(#2\right)\mathclose{}} \newcommand{\cinterval}[2]{\left[#1,#2\right]} \newcommand{\ointerval}[2]{\left(#1,#2\right)} \newcommand{\cointerval}[2]{\left[\left.#1,#2\right)\right.} \newcommand{\ocinterval}[2]{\left(\left.#1,#2\right]\right.} \newcommand{\point}[2]{\left(#1,#2\right)} \newcommand{\fd}[1]{#1'} \newcommand{\sd}[1]{#1''} \newcommand{\td}[1]{#1'''} \newcommand{\lz}[2]{\frac{d#1}{d#2}} \newcommand{\lzn}[3]{\frac{d^{#1}#2}{d#3^{#1}}} \newcommand{\lzo}[1]{\frac{d}{d#1}} \newcommand{\lzoo}[2]{{\frac{d}{d#1}}{\left(#2\right)}} \newcommand{\lzon}[2]{\frac{d^{#1}}{d#2^{#1}}} \newcommand{\lzoa}[3]{\left.{\frac{d#1}{d#2}}\right|_{#3}} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\sech}{\operatorname{sech}} \newcommand{\csch}{\operatorname{csch}} \newcommand{\lt}{ < } \newcommand{\gt}{ > } \newcommand{\amp}{ & } \)

Activity6.4Chain Rule with Leibniz Notation

So far we have worked with the chain rule as expressed using function notation. In some applications it is easier to think of the chain rule using Leibniz notation. Consider the following example.


During the 1990s, the amount of electricity used per day in Etown increased as a function of population at the rate of \(18\) kW/person. On July 1, 1997, the population of Etown was \(100{,}000\) and the population was decreasing at a rate of 6 people/day. In Equation (6.4.1) we use these values to determine the rate at which electrical usage was changing (with respect to time) in Etown on 7/1/1997. Please note that in this extremely simplified example we are ignoring all factors that contribute to citywide electrical usage other than population (such as temperature).\begin{equation}\left(18\frac{\text{kW}}{\text{person}}\right)\left(-6\frac{\text{people}}{\text{day}}\right)=-108\frac{\text{kW}}{\text{day}}\label{equation-leibniz-chain-example}\tag{6.4.1}\end{equation}Let's define \(\fe{g}{t}\) as the population of Etown \(t\) years after January 1, 1990 and \(\fe{f}{u}\) as the daily amount of electricity used in Etown when the population was \(u\). From the given information, \(\fe{g}{7.5}=100{,}000\), \(\fe{\fd{g}}{7.5}=-6\), and \(\fe{\fd{f}}{u}=18\) for all values of \(u\). If we let \(y=\fe{f}{u}\) where \(u=\fe{g}{t}\), then we have (from Equation (6.4.1)):\begin{equation}\left(\lzoa{y}{u}{u=100{,}000}\right)\left(\lzoa{u}{t}{t=7.5}\right)=\lzoa{y}{t}{t=7.5}\text{.}\label{equation-second-leibniz-chain-example}\tag{6.4.2}\end{equation}You should note that Equation (6.4.2) is an application of the chain rule expressed in Leibniz notation; specifically, the expression on the left side of the equal sign represents \(\fe{\fd{f}}{\fe{g}{7.5}}\fe{\fd{g}}{7.5}\).

In general, we could express the chain rule as shown in Equation (6.4.3).\begin{equation}\lz{y}{x}=\lz{y}{u}\cdot\lz{u}{x}\label{equation-leibniz-chain-rule}\tag{6.4.3}\end{equation}


Suppose that Carla is jogging in her sweet new running shoes. Suppose further that \(r=\fe{f}{t}\) is Carla's pace (mph) \(t\) hours after 1:00 pm and \(y=\fe{h}{r}\) is Carla's heart rate (bpm) when she jogs at a rate of \(r\) mph. In this context we can assume that all of Carla's motion was in one direction, so the words ‘speed’ and ‘velocity’ are completely interchangeable.


What is the meaning of \(\fe{f}{0.75}=7\)?


What is the meaning of \(\fe{h}{7}=125\)?


What is the meaning of \(\fe{\left(h\circ f\right)}{0.75}=125\)?


What is the meaning of \(\lzoa{r}{t}{t=0.75}=-0.00003\)?


What is the meaning of \(\lzoa{y}{r}{r=7}=8\)?


Assuming that all of the previous values are accurate, what is the value of \(\lzoa{y}{t}{t=0.75}\) and what does this value tell you about Carla?

Portions of SW 35th Avenue are extremely hilly. Suppose that you are riding your bike along SW 35th Ave from Vermont Street to Capitol Highway. Let \(u=\fe{d}{t}\) be the distance you have traveled (ft) where \(t\) is the number of seconds that have passed since you began your journey. Suppose that \(y=\fe{e}{u}\) is the elevation (m) of SW 35th Ave where \(u\) is the distance (ft) from Vermont St headed towards Capitol Highway.


What, including units, would be the meanings of \(\fe{d}{25}=300\), \(\fe{e}{300}=140\), and \(\fe{\left(e\circ d\right)}{25}=140\)?


What, including units, would be the meanings of \(\lzoa{u}{t}{t=25}=14\) and \(\lzoa{y}{u}{u=300}=-0.1\)?


Suppose that the values stated in Exercise are accurate. What, including unit, is the value of \(\lzoa{y}{t}{t=25}\)? What does this value tell you in the context of this problem?

According to Hooke's Law, the force (lb), \(F\), required to hold a spring in place when its displacement from the natural length of the spring is \(x\) (ft), is given by the formula \(F=kx\) where \(k\) is called the spring constant. The value of \(k\) varies from spring to spring.

Suppose that it requires \(120\) lb of force to hold a given spring \(1.5\) ft beyond its natural length.


Find the spring constant for this spring. Include units when substituting the values for \(F\) and \(x\) into Hooke's Law so that you know the unit on \(k\).


What, including unit, is the constant value of \(\lz{F}{x}\)?


Suppose that the spring is stretched at a constant rate of \(0.032\) fts. If we define \(t\) to be the amount of time (s) that passes since the stretching begins, what, including unit, is the constant value of \(\lz{x}{t}\)?


Use the chain rule to find the constant value (including unit) of \(\lz{F}{t}\). What is the contextual significance of this value?