Learning Outcomes
- Compute derivatives using the Product and Quotient Rules.
Section 2.4 The product and quotient rules (DF4)
Subsection 2.4.1 Activities
Activity 2.4.1.
Let \(f\) and \(g\) be the functions defined by
\begin{equation*}
f(t) = 2t^2 \, , \, g(t) = t^3 + 4t.
\end{equation*}
(a)
Find \(f'(t)\) and \(g'(t)\text{.}\)
(b)
Let \(P(t) = 2t^2 \, (t^3 + 4t)\) and observe that \(P(t) = f(t) \cdot g(t)\text{.}\) Rewrite the formula for \(P\) by distributing the \(2t^2\) term. Then, compute \(P'(t)\) using the power, sum, and scalar multiple rules.
(c)
True or false: \(P'(t) = f'(t) \cdot g'(t)\text{.}\)
Theorem 2.4.2. Product Rule.
If \(f\) and \(g\) are differentiable functions, then their product \(P(x) = f(x) \cdot g(x)\) is also a differentiable function, and
\begin{equation*}
P'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)\text{.}
\end{equation*}
Activity 2.4.3.
The product rule is a powerful tool, but sometimes it isn’t necessary; a more elementary rule may suffice. For which of the following functions can you find the derivative without using the product rule? Select all that apply.
- \(\displaystyle f(x)=e^x \sin x\)
- \(\displaystyle f(x)=\sqrt{x}(x^3+3x-3)\)
- \(\displaystyle f(x)=(4)(x^5)\)
- \(\displaystyle f(x)=x \ln x\)
Activity 2.4.4.
Find the derivative of the following functions using the product rule.
(a)
\(f(x)=(x^2+3x)\sin x\)(b)
\(f(x)=e^x \cos x\)(c)
\(f(x)=x^2\ln x\)Activity 2.4.5.
Let \(f\) and \(g\) be the functions defined by
\begin{equation*}
f(t) = 2t^2 \, , \, g(t) = t^3 + 4t.
\end{equation*}
(a)
(b)
Let \(Q(t) = \frac{t^3 + 4t}{2t^2}\) and observe that \(Q(t) = \frac{g(t)}{f(t)}\text{.}\) Rewrite the formula for \(Q\) by dividing each term in the numerator by the denominator and use rules of exponents to write \(Q\) as a sum of scalar multiples of power functions. Then, compute \(Q'(t)\) using the sum and scalar multiple rules.
(c)
True or false: \(Q'(t) = \frac{g'(t)}{f'(t)}\text{.}\)
Theorem 2.4.6. Quotient Rule.
If \(f\) and \(g\) are differentiable functions, then their quotient \(Q(x) = \frac{f(x)}{g(x)}\) is also a differentiable function for all \(x\) where \(g(x) \ne 0\) and
\begin{equation*}
Q'(x) = \frac{ f'(x) \cdot g(x) - f(x) \cdot g'(x)}{g(x)^2}\text{.}
\end{equation*}
Activity 2.4.7.
Just like with the product rule, there are times when we can find the derivative of a quotient using elementary rules rather than the quotient rule. For which of the following functions can you find the derivative without using the quotient rule? Select all that apply.
- \(\displaystyle f(x) = \frac{6}{x^3}\)
- \(\displaystyle f(x) = \frac{2}{\ln x}\)
- \(\displaystyle f(x) = \frac{e^x}{\sin x}\)
- \(\displaystyle f(x) = \frac{x^3+3x}{x}\)
Activity 2.4.8.
Find the derivative of the following functions using the quotient rule (or, if applicable, an elementary rule).
(a)
\(f(x) = \frac{6}{x^3}\)(b)
\(f(x) = \frac{2}{\ln x}\)(c)
\(f(x) = \frac{e^x}{\sin x}\)(d)
\(f(x) = \frac{x^3+3x}{x}\)Activity 2.4.9.
Demonstrate and explain how to find the derivative of the following functions. Be sure to explicitly denote which derivative rules (product, quotient, sum and difference, etc.) you are using in your work.
(a)
\begin{equation*}
f(w)= -\frac{3 \, w^{2} + 5 \, w - 2}{\sin\left(w\right)}
\end{equation*}
(b)
\begin{equation*}
g(t)= \frac{t^{2} + 6 \, t + 1}{t^{2}}
\end{equation*}
(c)
\begin{equation*}
h(t)= -2 \, {\left(t^{2} + 3 \, t + 3\right)} \cos\left(t\right)
\end{equation*}
Note 2.4.10.
We have found the derivatives of \(\sin x\) and \(\cos x\text{,}\) but what about the other trigonometric functions? It turns out that the quotient rule along with some trig identities can help us! (See Khan Academy 1 for a reminder of trig identities.)
Activity 2.4.11.
Consider the function \(f(x) = \tan x \text{,}\) and remember that \(\tan x = \frac{\sin x}{\cos x}\text{.}\)
(a)
What is the domain of \(f\text{?}\)
(b)
Use the quotient rule to show that one expression for \(f'(x)\) is
\begin{equation*}
f'(x) = \frac{(\cos x)(\cos x) + (\sin x)(\sin x)}{(\cos x)^2}\text{.}
\end{equation*}
(c)
Which trig identity might be useful here to simplify this expression? How can this identity be used to find a simpler form for \(f'(x)\text{?}\)
(d)
Recall that \(\sec x = \frac{1}{\cos x}\text{.}\) How can we express \(f'(x)\) in terms of the secant function?
(e)
For what values of \(x\) is \(f'(x)\) defined? How does this domain compare to the domain of \(f\text{?}\)
Activity 2.4.12.
Let \(g(x) = \cot x \text{,}\) and recall that \(\cot x = \frac{\cos x }{\sin x }\text{.}\)
(a)
What is the domain of \(g(x)\text{?}\)
(b)
Use the quotient rule to develop a formula for \(g'(x)\) that is expressed completely in terms of \(\sin x\) and \(\cos x\text{.}\)
(c)
Use other relationships among trigonometric functions to write \(g'(x)\) only in terms of the cosecant function.
(d)
What is the domain of \(g'(x)\text{?}\) How does this domain compare to the domain of \(g'(x)\text{?}\)
Activity 2.4.13.
Let \(h(x) = \sec x \text{,}\) and recall that \(\sec x = \frac{1}{\cos x }\text{.}\)
(a)
What is the domain of \(h(x)\text{?}\)
(b)
Use the quotient rule to develop a formula for \(h'(x)\) that is expressed completely in terms of \(\sin x\) and \(\cos x\text{.}\)
(c)
Use other relationships among trigonometric functions to write \(h'(x)\) only in terms of the the tangent and secant functions.
(d)
What is the domain of \(h'(x)\text{?}\) How does this domain compare to the domain of \(h'(x)\text{?}\)
Activity 2.4.14.
Let \(p(x) = \csc x \text{,}\) and recall that \(\csc x = \frac{1}{\sin x }\text{.}\)
(a)
What is the domain of \(p(x)\text{?}\)
(b)
Use the quotient rule to develop a formula for \(p'(x)\) that is expressed completely in terms of \(\sin x\) and \(\cos x\text{.}\)
(c)
Use other relationships among trigonometric functions to write \(h'(x)\) only in terms of the the cotangent and cosecant functions.
(d)
What is the domain of \(p'(x)\text{?}\) How does this domain compare to the domain of \(p'(x)\text{?}\)
Theorem 2.4.15.
We can now summarize the derivatives of all six trigonometric functions.
- \(\displaystyle \frac{d}{dx} \sin x = \cos x\)
- \(\displaystyle \frac{d}{dx} \cos x = -\sin x\)
- \(\displaystyle \frac{d}{dx} \tan x = (\sec x)^2\)
- \(\displaystyle \frac{d}{dx} \cot x = -(\csc x)^2\)
- \(\displaystyle \frac{d}{dx} \sec x = \sec x \tan x \)
- \(\displaystyle \frac{d}{dx} \csc x = -\csc x \cot x \)
Activity 2.4.16.
Consider the functions
\begin{equation*}
f(x)=3 \, \cos\left(x\right), \, \, g(x)=x^{2} + 3 \, e^{x}
\end{equation*}
and the function \(h(x)\) for which a table of values is given.
\begin{equation*}
\begin{array}{c|ccc}
x & -1& 0& 2 \\\hline
h(x) & -4& -1& 3 \\\hline
h'(x) & 0& -1& 1 \\
\end{array}
\end{equation*}
In answering the following questions, be sure to explicitly denote which derivative rules (product, quotient, sum/difference, etc.) you are using in your work.
(a)
Find the derivative of \(f(x)\cdot g(x)\text{.}\)
(b)
Find the derivative of \(\displaystyle \frac{f(x)}{g(x)}\text{.}\)
(c)
Find the value of the derivative of \(f(x) \cdot h(x) \) at \(x=-1\text{.}\)
(d)
Find the value of the derivative of \(\displaystyle \frac{g(x)}{h(x)}\) at \(x=0\text{.}\)
(e)
Consider the function
\begin{equation*}
r(x) = 3 \, \cos\left(x\right) \cdot x .
\end{equation*}
Find \(r'(x)\text{,}\) \(r''(x)\text{,}\) \(r'''(x)\text{,}\) and \(r^{(4)}(x)\) so the first, second, third, and fourth derivative of \(r(x)\text{.}\) What pattern do you notice? What do you expect the twelfth derivative of \(r(x)\) to be?
Activity 2.4.17.
(a)
Differentiate \(y = \displaystyle \frac{e^x}{x}, y = \displaystyle \frac{e^x}{x^2}, y = \displaystyle \frac{e^x}{x^3}\text{.}\) Simplify your answers as much as possible.
(b)
What do you expect the derivative of \(y = \displaystyle \frac{e^x}{x^n}\) to be? Prove your guess!
(c)
What do your answers above tell you above the shape of the graph of \(y = \displaystyle \frac{e^x}{x^n}\text{?}\) Study how the sign of the numerator and the denominator change in the first derivative to determine when the behavior changes!
Activity 2.4.18.
The quantity \(q\) of skateboards sold depends on the selling price \(p\) of a skateboard, so we write \(q=f(p)\text{.}\) You are given that
\begin{equation*}
f(140) = 15000, \, \, \, f'(140)= -100
\end{equation*}
(a)
What does the data provided tell you about the sales of skateboards?
(b)
The total revenue, \(R\text{,}\) earned by the sale of skateboards is given by \(R=q \cdot p = f(p) \cdot p\text{.}\) Explain why.
(c)
Find the derivative of the revenue when \(p=140\text{,}\) so find the value of
\begin{equation*}
\frac{dR}{dp}\Big|_{p=140}.
\end{equation*}
(d)
What is the sign of the quantity above? What do you think would happen to the revenue if the price was changed from $140 to $141?
Activity 2.4.19.
Let \(f(v)\) be the gas consumption in liters per kilometer (l/km) of a car going at velocity \(v\) kilometers per hour (km/hr). So if the car is going at velocity \(v\text{,}\) then \(f(v)\) tells you how many liters of gas the car uses to go one kilometer. You are given the following data
\begin{equation*}
f(50)=0.04, \, \, \, f'(50)=0.0004
\end{equation*}
(a)
Let \(g(v)\) be the distance (in kilometers) that the same car covers per liter of gas at velocity \(v\text{.}\) What are the units of the output of \(g(v)\text{?}\) Use these units to infer how to write \(g(v)\) in terms of \(f(v)\text{,}\) then find \(g(50)\) and \(g'(50)\text{.}\)
(b)
Let \(h(v)\) be the gas consumption over time, so the liters of gas consumed per hour by the same car going at velocity \(v\text{.}\) What are the units of the output of \(h(v)\text{?}\) Use these units to infer how to write \(h(v)\) in terms of \(f(v)\text{,}\) then find \(h(50)\) and \(h'(50)\text{.}\)
(c)
How would you explain the practical meaning of your findings to a driver who knows no calculus?
