# Calculus for Biological Scientists

## Section2.7Derivatives of Products and Quotients

So far, we can differentiate power functions ($$x^n$$), exponential functions ($$b^x$$), and the two fundamental trigonometric functions ($$\sin(x)$$ and $$\cos(x)$$). With the sum rule and constant multiple rules, we can also compute the derivative of combined functions.

### Example2.7.1.

Differentiate
\begin{equation*} f(x) = 7x^{11} - 4 \cdot 9^x + \pi \sin(x) - \sqrt{3}\cos(x) \end{equation*}
Because $$f$$ is a sum of basic functions, we can now quickly say that $$f'(x) = 77x^{10} - 4 \cdot 9^x \ln(9) + \pi \cos(x) + \sqrt{3} \sin(x)\text{.}$$
What about a product or quotient of two basic functions, such as
\begin{equation*} p(z) = z^3 \cos(z)\text{,} \end{equation*}
or
\begin{equation*} q(t) = \frac{\sin(t)}{2^t}\text{?} \end{equation*}
While the derivative of a sum is the sum of the derivatives, it turns out that the rules for computing derivatives of products and quotients are more complicated.

### Warm-Up2.7.1.

Let $$f$$ and $$g$$ be the functions defined by $$f(t) = 2t^2$$ and $$g(t) = t^3 + 4t\text{.}$$
1. Determine $$f'(t)$$ and $$g'(t)\text{.}$$
2. 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 sum and constant multiple rules.
3. True or false: $$p'(t) = f'(t) \cdot g'(t)\text{.}$$
4. 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 simplify to write $$q$$ as a sum of constant multiples of powers of $$t\text{.}$$ Then, compute $$q'(t)$$ using the sum and constant multiple rules.
5. True or false: $$q'(t) = \frac{g'(t)}{f'(t)}\text{.}$$

### Subsection2.7.1The product rule

As part (b) of Warm-Up 2.7.1 shows, it is not true in general that the derivative of a product of two functions is the product of the derivatives of those functions. To see why this is the case, we consider an example involving meaningful functions.
Say that we are tracking carbon emissions for a particular population over time. Let $$N(t)$$ represent the number of individuals on day $$t\text{,}$$ where $$t = 0$$ represents the first day on which we begin tracking. Let $$S(t)$$ give the amount of carbon emissions from one individual on day $$t\text{;}$$ the units on $$S(t)$$ are pounds per person (the rate of carbon emissions). To compute the total carbon emissions for the whole population on day $$t\text{,}$$ we take the product
\begin{equation*} V(t) = N(t) \, \text{individuals} \cdot S(t) \, \text{pounds per individual}\text{.} \end{equation*}
Observe that over time, both the number of individuals and the rate of carbon emissions will vary. The derivative $$N'(t)$$ measures the rate at which the number of individuals is changing, while $$S'(t)$$ measures the rate at which the carbon emission per individual is changing. How do these respective rates of change affect the rate of change of the total carbon emission function?
To help us understand the relationship among changes in $$N\text{,}$$ $$S\text{,}$$ and $$V\text{,}$$ let’s consider some specific data.
• Suppose that on day 100, there are 520 individuals and the rate of carbon emission is 2.75 pounds per individual. This tells us that $$N(100) = 520$$ and $$S(100) = 2.75\text{.}$$
• On day 100, the population grows by an additional 12 individuals (so the number of individuals is rising at a rate of 12 individuals per day).
• On that same day the rate of carbon emissions is rising at a rate of 0.75 pounds per individual per day.
In calculus notation, the latter two facts tell us that $$N'(100) = 12$$ (individuals per day) and $$S'(100) = 0.75$$ (pounds per individual per day). At what rate is the total carbon emission changing on day 100?
Observe that the increase in total value comes from two sources: the growing number of individuals, and the rising rate of carbon emissions per individual. If only the number of individuals is increasing (and the value of the rate of carbon emission is constant), the rate at which total carbon emissions would rise is the product of the current rate of carbon emission and the rate at which the number of individuals is changing. That is, the rate at which total carbon emissions would change is given by
\begin{equation*} S(100) \cdot N'(100) = 2.75 \, \frac{\text{pounds} }{\text{individual} } \cdot 12 \, \frac{\text{individuals} }{\text{day} } = 33 \, \frac{\text{pounds} }{\text{day} }\text{.} \end{equation*}
Note particularly how the units make sense and show the rate at which the total value $$V$$ is changing, measured in pounds per day.
If instead the number of individuals is constant, but the rate of carbon emissions is rising, the rate at which the total carbon emission would rise is the product of the number of individuals and the rate of change of the carbon emission rate. The total carbon emission is rising at a rate of
\begin{equation*} N(100) \cdot S'(100) = 520 \, \text{individuals} \cdot 0.75 \, \frac{\text{pounds per individual} }{\text{day} } = 390 \, \frac{\text{pounds} }{\text{day} }\text{.} \end{equation*}
Of course, when both the number of individuals and the rate of carbon emissions are changing, we have to include both of these sources. In that case the rate at which the total carbon emission is rising is
\begin{equation*} V'(100) = S(100) \cdot N'(100) + N(100) \cdot S'(100) = 33 + 390 = 423 \, \frac{\text{pounds} }{\text{day} }\text{.} \end{equation*}
We expect the total carbon emission from the population to rise by about 423 pounds on the 100th day. 1
Next, we expand our perspective from the specific example above to the more general and abstract setting of a product $$p$$ of two differentiable functions, $$f$$ and $$g\text{.}$$ If $$P(x) = f(x) \cdot g(x)\text{,}$$ our work above suggests that $$P'(x) = f'(x) g(x) + f(x)g'(x)\text{.}$$ Indeed, a formal proof using the limit definition of the derivative can be given to show that the following rule, called the product rule, holds in general.

#### 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) g(x) + f(x)g'(x) \text{.} \end{equation*}
In light of the earlier example involving carbon emissions, the product rule also makes sense intuitively: the rate of change of $$P$$ should take into account both how fast $$f$$ and $$g$$ are changing, as well as how large $$f$$ and $$g$$ are at the point of interest. In words the product rule says: if $$P$$ is the product of two functions $$f$$ (the first function) and $$g$$ (the second), then “the derivative of $$P$$ is the derivative of the first times the second, plus the first times the derivative of the second.” It is often a helpful mental exercise to say this phrasing aloud when executing the product rule.

#### Example2.7.2.

If $$P(z) = z^3 \cdot \cos(z)\text{,}$$ we can use the product rule to differentiate $$P\text{.}$$ The first function is $$z^3$$ and the second function is $$\cos(z)\text{.}$$ By the product rule, $$P'$$ will be given by the derivative of the first, $$3z^2\text{,}$$ times the second, $$\cos(z)\text{,}$$ plus the first, $$z^3\text{,}$$ times the derivative of the second, $$-\sin(z)\text{.}$$ That is,
\begin{equation*} P'(z) = 3z^2 \cos(z)+ z^3(-\sin(z)) = 3z^2 \cos(z)-z^3 \sin(z) \text{.} \end{equation*}

#### Activity2.7.2.

Use the product rule to answer each of the questions below. Throughout, be sure to carefully label any derivative you find by name. It is not necessary to algebraically simplify any of the derivatives you compute.
1. Let $$m(w)=3w^{17} 4^w\text{.}$$ Find $$m'(w)\text{.}$$
2. Let $$h(t) = (\sin(t) + \cos(t))t^4\text{.}$$ Find $$h'(t)\text{.}$$
3. Determine the slope of the tangent line to the curve $$y = f(x)$$ at the point where $$a = 1$$ if $$f$$ is given by the rule $$f(x) = e^x \sin(x)\text{.}$$
4. Find the equation of the tangent line to the function $$y = g(x)$$ at the point where $$a = -1$$ if $$g$$ is given by the rule $$g(x) = (x^2 + x) 2^x\text{.}$$

### Subsection2.7.2The quotient rule

Because quotients and products are closely linked, we can use the product rule to understand how to take the derivative of a quotient. Let $$Q(x)$$ be defined by $$Q(x) = f(x)/g(x)\text{,}$$ where $$f$$ and $$g$$ are both differentiable functions. It turns out that $$Q$$ is differentiable everywhere that $$g(x) \ne 0\text{.}$$ We would like a formula for $$Q'$$ in terms of $$f\text{,}$$ $$g\text{,}$$ $$f'\text{,}$$ and $$g'\text{.}$$ Multiplying both sides of the formula $$Q = f/g$$ by $$g\text{,}$$ we observe that
\begin{equation*} f(x) = Q(x) \cdot g(x)\text{.} \end{equation*}
Now we can use the product rule to differentiate $$f\text{.}$$
\begin{equation*} f'(x) = Q'(x)g(x) + Q(x) g'(x)\text{.} \end{equation*}
We want to know a formula for $$Q'\text{,}$$ so we solve this equation for $$Q'(x)\text{.}$$
\begin{equation*} Q'(x) g(x) = f'(x) - Q(x) g'(x) \end{equation*}
and dividing both sides by $$g(x)\text{,}$$ we have
\begin{equation*} Q'(x) = \frac{f'(x) - Q(x) g'(x)}{g(x)}\text{.} \end{equation*}
Finally, we recall that $$Q(x) = \frac{f(x)}{g(x)}\text{.}$$ Substituting this expression in the preceding equation, we have
\begin{align*} Q'(x) =\mathstrut \amp \frac{f'(x) - \frac{f(x)}{g(x)} g'(x)}{g(x)}\\ =\mathstrut \amp \frac{f'(x) - \frac{f(x)}{g(x)} g'(x)}{g(x)} \cdot \frac{g(x)}{g(x)}\\ =\mathstrut \amp \frac{ f'(x)g(x) - f(x) g'(x)}{g(x)^2}\text{.} \end{align*}
This calculation gives us the quotient rule.

#### 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)g(x) - f(x) g'(x)}{g(x)^2}\text{.} \end{equation*}
As with the product rule, it can be helpful to think of the quotient rule verbally. If a function $$Q$$ is the quotient of a top function $$f$$ and a bottom function $$g\text{,}$$ then $$Q'$$ is given by “the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared.” Note that remembering the order of the product rule given in the previous subsection can help in remembering the correct order for the numerator of the quotient rule.

#### Example2.7.3.

If $$Q(t) = \sin(t)/2^t\text{,}$$ we call $$\sin(t)$$ the top function and $$2^t$$ the bottom function. By the quotient rule, $$Q'$$ is given by the derivative of the top, $$\cos(t)$$ times the bottom, $$2^t\text{,}$$ minus the top, $$\sin(t)\text{,}$$ times the derivative of the bottom, $$2^t \ln(2)\text{,}$$ all over the bottom squared, $$(2^t)^2\text{.}$$ That is,
\begin{equation*} Q'(t) = \frac{ \cos(t)2^t - \sin(t) 2^t \ln(2)}{(2^t)^2}\text{.} \end{equation*}
In this particular example, it is possible to simplify $$Q'(t)$$ by removing a factor of $$2^t$$ from both the numerator and denominator, so that
\begin{equation*} Q'(t) = \frac{\cos(t) - \sin(t) \ln(2)}{2^t}\text{.} \end{equation*}
In general, we must be careful in doing any such simplification, as we don’t want to execute the quotient rule correctly but then make an algebra error.

#### Activity2.7.3.

Use the quotient rule to answer each of the questions below. Throughout, be sure to carefully label any derivative you find by name. That is, if you’re given a formula for $$f(x)\text{,}$$ clearly label the formula you find for $$f'(x)\text{.}$$ It is not necessary to algebraically simplify any of the derivatives you compute.
1. Let $$r(z)=\frac{3^z}{z^4 + 1}\text{.}$$ Find $$r'(z)\text{.}$$
2. Let $$v(t) = \frac{\sin(t)}{\cos(t) + t^2}\text{.}$$ Find $$v'(t)\text{.}$$
3. Determine the slope of the tangent line to the curve $$\displaystyle R(x) = \frac{x^2 - 2x - 8}{x^2 - 9}$$ at the point where $$x = 0\text{.}$$
4. When a camera flashes, the intensity $$I$$ of light seen by the eye is given by the function
\begin{equation*} I(t) = \frac{100t}{e^t}\text{,} \end{equation*}
where $$I$$ is measured in candles and $$t$$ is measured in milliseconds. Compute $$I'(0.5)\text{,}$$ $$I'(2)\text{,}$$ and $$I'(5)\text{;}$$ include appropriate units on each value; and discuss the meaning of each.

### Subsection2.7.3Combining rules

In order to apply the derivative shortcut rules correctly we must recognize the fundamental structure of a function.

#### Example2.7.4.

Determine the derivative of the function
\begin{equation*} f(x) = x\sin(x) + \frac{x^2}{\cos(x) + 2}\text{.} \end{equation*}
How do we decide which rules to apply? Our first task is to recognize the structure of the function. This function $$f$$ is a sum of two slightly less complicated functions, so we can apply the sum rule 2  to get
\begin{align*} f'(x) =\mathstrut \amp \frac{d}{dx} \left[ x\sin(x) + \frac{x^2}{\cos(x) + 2} \right]\\ =\mathstrut \amp \frac{d}{dx} \left[ x\sin(x) \right] + \frac{d}{dx}\left[ \frac{x^2}{\cos(x) + 2} \right] \end{align*}
Now, the left-hand term above is a product, so the product rule is needed there, while the right-hand term is a quotient, so the quotient rule is required. Applying these rules respectively, we find that
\begin{align*} f'(x) =\mathstrut \amp \left( x \cos(x) + \sin(x) \right) + \frac{(\cos(x) + 2) 2x - x^2(-\sin(x))}{(\cos(x) + 2)^2}\\ =\mathstrut \amp x \cos(x) + \sin(x) + \frac{2x\cos(x) + 4x + x^2\sin(x)}{(\cos(x) + 2)^2}\text{.} \end{align*}

#### Example2.7.5.

Differentiate
\begin{equation*} s(y) = \frac{y \cdot 7^y}{y^2 + 1}\text{.} \end{equation*}
The function $$s$$ is a quotient of two simpler functions, so the quotient rule will be needed. To begin, we set up the quotient rule and use the notation $$\frac{d}{dy}$$ to indicate the derivatives of the numerator and denominator. Thus,
\begin{equation*} s'(y) = \frac{\frac{d}{dy}\left[ y \cdot 7^y \right] \cdot(y^2 + 1) - y \cdot 7^y \cdot \frac{d}{dy}\left[y^2 + 1 \right]}{(y^2 + 1)^2}\text{.} \end{equation*}
Now, there remain two derivatives to calculate. The first one, $$\frac{d}{dy}\left[ y \cdot 7^y \right]$$ calls for use of the product rule, while the second, $$\frac{d}{dy}\left[y^2 + 1 \right]$$ needs only the sum rule. Applying these rules, we now have
\begin{equation*} s'(y) = \frac{[1 \cdot 7^y + y \cdot 7^y \ln(7) ](y^2 + 1) - y \cdot 7^y [2y]}{(y^2 + 1)^2}\text{.} \end{equation*}
While some simplification is possible, we are content to leave $$s'(y)$$ in its current form.
Success in applying derivative rules begins with recognizing the structure of the function, followed by the careful and diligent application of the relevant derivative rules. The best way to become proficient at this process is to do a large number of examples.

#### Activity2.7.4.

Use relevant derivative rules to answer each of the questions below. Throughout, be sure to use proper notation and carefully label any derivative you find by name.
1. Let $$f(r) = (5r^3 + \sin(r))(4^r - 2\cos(r))\text{.}$$ Find $$f'(r)\text{.}$$
2. Let $$\displaystyle p(t) = \frac{\cos(t)}{t^6 \cdot 6^t}\text{.}$$ Find $$p'(t)\text{.}$$
3. Let $$g(z) = 3z^7 e^z - 2z^2 \sin(z) + \frac{z}{z^2 + 1}\text{.}$$ Find $$g'(z)\text{.}$$
4. A moving particle has its position in feet at time $$t$$ in seconds given by the function $$s(t) = \frac{3\cos(t) - \sin(t)}{e^t}\text{.}$$ Find the particle’s instantaneous velocity at the moment $$t = 1\text{.}$$
5. Suppose that $$f(x)$$ and $$g(x)$$ are differentiable functions and it is known that $$f(3) = -2\text{,}$$ $$f'(3) = 7\text{,}$$ $$g(3) = 4\text{,}$$ and $$g'(3) = -1\text{.}$$ If $$p(x) = f(x) \cdot g(x)$$ and $$\displaystyle q(x) = \frac{f(x)}{g(x)}\text{,}$$ calculate $$p'(3)$$ and $$q'(3)\text{.}$$
As the algebraic complexity of the functions we are able to differentiate continues to increase, it is important to remember that all of the derivative’s meaning continues to hold. Regardless of the structure of the function $$f\text{,}$$ the value of $$f'(a)$$ tells us the instantaneous rate of change of $$f$$ with respect to $$x$$ at the moment $$x = a\text{,}$$ as well as the slope of the tangent line to $$y = f(x)$$ at the point $$(a,f(a))\text{.}$$

### Subsection2.7.4Summary

• #### Question2.7.6.

How does the algebraic structure of a function guide us in computing its derivative using shortcut rules?
If a function is a sum, product, or quotient of simpler functions, then we can use the sum, product, or quotient rules to differentiate it in terms of the simpler functions and their derivatives.
• #### Question2.7.7.

How do we compute the derivative of a product of two basic functions in terms of the derivatives of the basic functions?
The product rule tells us that if $$P$$ is a product of differentiable functions $$f$$ and $$g$$ according to the rule $$P(x) = f(x) g(x)\text{,}$$ then
\begin{equation*} P'(x) = f'(x)g(x)+ f(x)g'(x) \text{.} \end{equation*}
• #### Question2.7.8.

How do we compute the derivative of a quotient of two basic functions in terms of the derivatives of the basic functions?
The quotient rule tells us that if $$Q$$ is a quotient of differentiable functions $$f$$ and $$g$$ according to the rule $$Q(x) = \frac{f(x)}{g(x)}\text{,}$$ then
\begin{equation*} Q'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\text{.} \end{equation*}
• #### Question2.7.9.

How do the product and quotient rules combine with the sum and constant multiple rules to expand the library of functions we can differentiate quickly?
Along with the constant multiple and sum rules, the product and quotient rules enable us to compute the derivative of any function that consists of sums, constant multiples, products, and quotients of basic functions. For instance, if $$F$$ has the form
\begin{equation*} F(x) = \frac{2a(x) - 5b(x)}{c(x) \cdot d(x)}\text{,} \end{equation*}
then $$F$$ is a quotient, in which the numerator is a sum of constant multiples and the denominator is a product. Thus, the derivative of $$F$$ can be found by applying the quotient rule and then using the sum and constant multiple rules to differentiate the numerator and the product rule to differentiate the denominator.

### Exercises2.7.5Exercises

#### 1.

Show that $$\dfrac{d}{dx}\left[ \tan(x)\right] = \sec^2(x)\text{.}$$
HINT: Recall that $$\tan(x) = \dfrac{\sin(x)}{\cos(x)}$$ and $$\sec(x)= \dfrac{1}{\cos(x)}\text{.}$$

#### 2.

Let $$f$$ and $$g$$ be differentiable functions for which the following information is known: $$f(2) = 5\text{,}$$ $$g(2) = -3\text{,}$$ $$f'(2) = -1/2\text{,}$$ $$g'(2) = 2\text{.}$$
1. Let $$h$$ be the new function defined by the rule $$h(x) = g(x) \cdot f(x)\text{.}$$ Determine $$h(2)$$ and $$h'(2)\text{.}$$
2. Find an equation for the tangent line to $$y = h(x)$$ at the point $$(2,h(2))$$ (where $$h$$ is the function defined in (a)).
3. Let $$r$$ be the function defined by the rule $$r(x) = \frac{g(x)}{f(x)}\text{.}$$ Is $$r$$ increasing, decreasing, or neither at $$a = 2\text{?}$$ Why?
4. Find an equation for the tangent line to $$y=r(x)$$ at the point $$(2,r(2))$$ (where $$r$$ is the function defined in (c)).

#### 3.

Let functions $$p$$ and $$q$$ be the piecewise linear functions given by their respective graphs in Figure 2.7.10. Use the graphs to answer the following questions.
1. Let $$r(x) = p(x) \cdot q(x)\text{.}$$ Determine $$r'(-2)$$ and $$r'(0)\text{.}$$
2. Find an equation for the tangent line to $$y = r(x)$$ at the point $$(2,r(2))\text{.}$$
3. Let $$z(x) = \frac{q(x)}{p(x)}\text{.}$$ Determine $$z'(0)$$ and $$z'(2)\text{.}$$
4. Let $$\ell(x) = p(x) \cdot e^x\text{.}$$ Determine $$\ell'(0)\text{.}$$

#### 4.

A farmer with large land holdings has historically grown a wide variety of crops. With the price of ethanol fuel rising, he decides that it would be prudent to devote more and more of his acreage to producing corn. As he grows more and more corn, he learns efficiencies that increase his yield per acre. In the present year, he used 7000 acres of his land to grow corn, and that land had an average yield of 170 bushels per acre. At the current time, he plans to increase his number of acres devoted to growing corn at a rate of 600 acres/year, and he expects that right now his average yield is increasing at a rate of 8 bushels per acre per year. Use this information to answer the following questions.
1. Say that the present year is $$t = 0\text{,}$$ that $$A(t)$$ denotes the number of acres the farmer devotes to growing corn in year $$t\text{,}$$ $$Y(t)$$ represents the average yield in year $$t$$ (measured in bushels per acre), and $$C(t)$$ is the total number of bushels of corn the farmer produces. What is the formula for $$C(t)$$ in terms of $$A(t)$$ and $$Y(t)\text{?}$$ Why?
2. What is the value of $$C(0)\text{?}$$ What does it measure?
3. Write an expression for $$C'(t)$$ in terms of $$A(t)\text{,}$$ $$A'(t)\text{,}$$ $$Y(t)\text{,}$$ and $$Y'(t)\text{.}$$ Explain your thinking.
4. What is the value of $$C'(0)\text{?}$$ What does it measure?
5. Based on the given information and your work above, write the equation of the tangent line to $$C(t)$$ at the point $$(0,C(0))\text{.}$$

#### 5.

You will find unit conversions (Section 1.3) to be especially helpful in answering the following questions.
Let $$f(v)$$ be the gas consumption (in liters/km) of a car going at velocity $$v$$ (in km/hour). In other words, $$f(v)$$ tells you how many liters of gas the car uses to go one kilometer if it is traveling at $$v$$ kilometers per hour. In addition, suppose that $$f(80)=0.05$$ and $$f'(80) = 0.0004\text{.}$$
1. Let $$g(v)$$ be the distance the same car goes on one liter of gas at velocity $$v\text{.}$$ What is the relationship between $$f(v)$$ and $$g(v)\text{?}$$ Hence find $$g(80)$$ and $$g'(80)\text{.}$$
2. Let $$h(v)$$ be the gas consumption in liters per hour of a car going at velocity $$v\text{.}$$ In other words, $$h(v)$$ tells you how many liters of gas the car uses in one hour if it is going at velocity $$v\text{.}$$ What is the algebraic relationship between $$h(v)$$ and $$f(v)\text{?}$$ Hence find $$h(80)$$ and $$h'(80)\text{.}$$
3. How would you explain the practical meaning of these function and derivative values to a driver who knows no calculus? Include units on each of the function and derivative values you discuss in your response.
While this example highlights why the product rule is true, there are some subtle issues to recognize. For one, if the rate of carbon emissions really does rise exactly 0.75 pounds per individual on day 100, and the number of individuals really rises by 12 on day 100, then we’d expect that $$V(101) = N(101) \cdot S(101) = 532 \cdot 3.5 = 1862\text{.}$$ If, as noted above, we expect the total carbon emission to rise by 423 pounds, then with $$V(100) = N(100) \cdot S(100) = 520 \cdot 2.75 = 1430\text{,}$$ then it seems we should find that $$V(101) = V(100) + 423 = 1853\text{.}$$ Why do the two results differ by 9? One way to understand why this difference occurs is to recognize that $$N'(100) = 12$$ represents an instantaneous rate of change, while our (informal) discussion has also thought of this number as the total change in the number of individuals over the course of a single day. The formal proof of the product rule reconciles this issue by taking the limit as the change in the input tends to zero.
When taking a derivative that involves the use of multiple derivative rules, it is often helpful to use the notation $$\frac{d}{dx} \left[ ~~\right]$$ to wait to apply subsequent rules. This is demonstrated in each of the two examples presented here.