3.3: The derivative function, F'

Definition 3.3.1: The function, F'

Suppose \(F\) is a function and for some number, \(x\), in its domain the rate of change of \(F\) exists at \(x\). Then the function \(F'\) (read ’F prime’) is defined by

\[F^{\prime}(x)=\text { the rate of change of } F \text { at } x \label{3.20}\]

for all numbers \(x\) in the domain of \(F\) for which the rate of change of \(F\) at \(x\) exists.

Equivalently,

\[F^{\prime}(x)=\text { the slope of the graph of } F \text { at }(x, F(x)) \label{3.21}\]

for points \(( x, F(x) )\) of the graph of F for which the tangent exists.

Symbolically, we may write

\[F^{\prime}(x)=\lim _{b \rightarrow x} \frac{F(b)-F(x)}{b-x} . \label{3.22}\]

The function \(F^{\prime}\) is called the derivative ⁴ of \(F\) (the function derived from \(F\)). When the independent variable of \(F\) is expressed as in \(F(x), F^{\prime} (x)\) is the derivative of \(F(x)\) with respect to \(x\).

Note

The derivative is one-half of calculus, perhaps 4 percent of your university education, and requires your attention. The next 200? pages of this text present biological and physical interpretations of the derivative, formulas for computing the derivatives of commonly encountered functions, and uses of derivatives in writing equations for mathematical models of biological and physical systems.

Example 3.3.1 You found in Chapter 1, Problem 1.9.1 on page 45 that light intensity, \(L\), at a distance, \(d\), from a linear slit of light was

\[L(d)=\frac{1.45}{d} \quad 3 \leq d \leq 16\]

where \(L\) was measured in \(mW/cm^{2}\) and \(d\) was measured in cm.

At what rate is the light decreasing when \(d\) = 5 cm? We find the derivative of \(L\). For any value of d between 3 and 16 cm,

\[\begin{aligned}
L^{\prime}(d) &=\lim _{b \rightarrow d} \frac{L(b)-L(d) \quad \mathrm{mW} / \mathrm{cm}^{2}}{b-d \mathrm{~cm}} \\
&=\lim _{b \rightarrow d} \frac{\frac{1.45}{b}-\frac{1.45}{d}}{b-d} \quad \frac{\mathrm{mW} / \mathrm{cm}^{2}}{\mathrm{~cm}} \\
&=\lim _{d \rightarrow 5}\left(-1.45 \frac{1}{d \times b}\right) \quad \frac{\mathrm{mW} / \mathrm{cm}^{2}}{\mathrm{~cm}} \\
&=\frac{-1.45}{d^{2}} \frac{\mathrm{mW} / \mathrm{cm}^{2}}{\mathrm{~cm}} \\
\text { For } d=5, \quad L^{\prime}(5) &=\left.\frac{-1.45}{d^{2}}\right|_{d=5}=\frac{-1.45}{5^{2}}=-0.058 \quad \frac{\mathrm{mW} / \mathrm{cm}^{2}}{\mathrm{~cm}}
\end{aligned}\]

We have just used a helpful notation: \(\left.\right|_{x=a}\)

For any function, G, \(\left.G(x)\right|_{x=a}=G(a) \text { . }\)

It is useful to compare the algebraic forms of \(L\) and \(L^{\prime}\),

\[L(d)=\frac{k}{d} \quad \text { and } \quad L^{\prime}(d)=-k \frac{1}{d^{2}}, \quad \text { and note that } \quad L^{\prime}(d)=-\frac{1}{k} L^{2}(d) .\]

Explore 3.3.1 For one mole of oxygen at \(300^{\circ}\) Kelvin the pressure, P in atmospheres, and volume, V in liters, are related by

\[P=\frac{1 \times 0.0820 \times 300}{V}\]

1. Find the derivative of P with respect to V .
2. What is \(P ^{\prime} (1)\)?

The question of units on the rate of change is important. We have used the following.

Units on \(F ^{\prime}\)

If \(\lim _{x \rightarrow a} G(x)=L\) then the units on L are the units on G.

It follows that the units on

\[F^{\prime}(x)=\lim _{b \rightarrow x} \frac{F(b)-F(x)}{b-x}\]

are the units on \(F\) divided by the the units on the domain of \(F\) (the units on \(x\)).

• If \(F(t)\) is distance in meters and \(t\) is time in seconds, then \(F ^{\prime} (t)\) is meters per second, or velocity.
• If \(F(x)\) is pressure in atmospheres and \(x\) is altitude in km, then \(F ^{\prime} (x)\) is commonly called the pressure gradient and is measured in atm/km.
• If \(F(t)\) is population size in individuals and \(t\) is time in years, then \(F ^{\prime} (t)\) is population growth rate (which might be negative) in individuals per year.

It is useful to see the rate of change of \(F\) over the whole domain of \(F\). In Figure 3.3.1A is the graph of a function, called a logistic function, that is typical of the size of population that starts at low density and grows in a limited environment. At time \(t = 0\) the population size is \(P(0) = 1\) and the maximum supportable population is \(M = 10\). Its derivative is shown in 3.3.1B and illustrates population growth rate. The maximum of the derivative occurs at \(t \doteq 3.17\)and this marks the steepest part of the population graph. The growth rate initially is low, rises to a maximum at \(t \doteq 3.17\), and decreases again as population density nears its maximum.

Figure \(\PageIndex{1}\): A. Graph of a classical logistic curve, \(L\), describing population size as a function of time. B. A graph of \(L ^{\prime}\) , the population growth rate.

Figure \(\PageIndex{2}\): The graph of a function \(P\) and its derivative \(P ^{\prime}\) .

Example 3.3.2 Problem. The graphs of a function \(P\) and its derivative \(P ^{\prime}\) are shown in Figure 3.3.2. Which graph is the graph of \(P\)?

Solution. We claim that graph 1 is not \(P\), because every tangent to graph 1 has negative slope and some y-coordinates of graph 2 are positive. Therefore graph 2 must be the graph of \(P\).

Alternate notation. Calculus originated in England with Sir Isaac Newton (1642-1727) and in Germany with Gottfried Wilhelm Leibniz (1646-1716), and indeed some elements of it were anticipated by the Greek mathematicians. Given the multiple origins and a 300 year history, it is not surprising that there are several notations for derivative. Newton used \(y ^{\prime}\) for the ‘fluxion’ of \(y\) (rate of change of \(y\)). But when the independent variable was time, Newton used the symbol \(\dot{y}\) for the rate of change of \(y\) and \(\ddot{y}\) for the second derivative of \(y\) (the derivative of the derivative of \(y\)). Therefore if \(y\) denotes distance, \(\dot{y}\) denotes velocity and \(\ddot{y}\) denotes acceleration. The most common notation, \(\frac{d F}{d t}\) , for the derivative was introduced and used by Leibnitz. When discussing the rate of change of \(F\) at a number \(a\) one may see

\[F^{\prime}(a) \quad \text { or }\left.\quad \frac{d F}{d x}\right|_{x=a} \quad \text { or } \quad \frac{d F}{d x} \text { or } \dot{F} \text { or } \dot{F}(a)\]

A function that has a derivative is called a differentiable function. To differentiate \(F\) means to compute the derivative, \(F ^{\prime}\) . “Differentiable” comes from the concept of a ‘differential’ which is a cousin of an elusive concept called ‘infinitesimal’. An infinitesimal is a positive number that is less than all other positive numbers, which is possible only in an extension of our familiar number system. An infinitesimal change, \(dx\), in \(x\) causes an infinitesimal change, \(dF\), in \(F(x)\) and the derivative is the ratio \(\frac{d F}{d t}\). The concept of a limit is considered to be less mysterious than is infinitesimal, and can easily be put on a quite sound footing, whereas it is difficult to define ‘infinitesimal’ clearly ⁵.

It is sometimes preferable to substitute \(h\) for \(b − x\) so that \(b = x + h\), and write

\[F^{\prime}(x)=\lim _{b \rightarrow x} \frac{F(b)-F(x)}{b-x} \quad \text { as } \quad F^{\prime}(x)=\lim _{h \rightarrow 0} \frac{F(x+h)-F(x)}{h}\]

 

Exercises for Section 3.3, The derivative function, F'.

Exercise 3.3.1 Use Equation \ref{3.22},

\[F^{\prime}(x)=\lim _{b \rightarrow x} \frac{F(b)-F(x)}{b-x}\]

to compute \(F ^{\prime} (x)\) for

1. \(\quad F(x)=x^{2}\)
2. \(\quad F(x)=2 x^{2}\)
3. \(\quad F(x)=x^{2}+1\)
4. \(\quad F(x)=x^{3}\)
5. \(\quad F(x)=4 x^{3}\)
6. \(\quad F(x)=x^{3}-1\)
7. \(\quad F(x)=x^{2}+x\)
8. \(\quad F(x)=x^{2}+x^{3}\)
9. \(\quad F(x)=3 x+1\)
10. \(\quad F(x)=\sqrt{x}\)
11. \(\quad F(x)=4 \sqrt{x}\)
12. \(\quad F(x)=4+\sqrt{x}\)
13. \(\quad F(x)=5\)
14. \(\quad F(x)=\frac{1}{x}\)
15. \(\quad F(x)=5+\frac{1}{x}\)
16. \(\quad F(x)=\frac{1}{x^{2}}\)
17. \(\quad F(x)=\frac{5}{x^{2}}\)
18. \(\quad F(x)=5+\frac{1}{x^{2}}\)

Exercise 3.3.2 Use the equation,

\[F^{\prime}(x)=\lim _{h \rightarrow 0} \frac{F(x+h)-F(x)}{h}\]

to comupute \(F ^{\prime} (x)\) for:

1. \(\quad F(x)=x^{2}\)
2. \(\quad F(x)=3 x^{2}\)
3. \(\quad F(x)=x^{2}+5\)
4. \(\quad F(x)=x^{-1}\)
5. \(\quad F(x)=2 x^{-1}\)
6. \(\quad F(x)=x^{-1}-7\)

Exercise 3.3.3 In Figure Ex. 3.3.3A and Figure Ex. 3.3.3B are four pairs of graphs of a function \(P\) and and its derivative, \(P ^{\prime}\). For each pair, which is the graph of \(P\)? Explain your choice.

Figure for Exercise 3.3.3 A. Graphs of a function and its derivative for Exercise 3.3.3.

Figure for Exercise 3.3.3 B. Graphs of a function and its derivative for Exercise 3.3.3.

Exercise 3.3.4

a. In Exercise Figure 3.3.4A is the graph of a function, \(F\). Estimate the slopes of the tangents to that graph at the points marked on the graph. Plot a new graph of slopes vs the dependent variable for \(F\). Sketch a graph of \(F ^{\prime}\) .

b. Repeat the steps of the part a. for the graph of a function, \(G\), in Exercise Figure 3.3.4B.

Figure for Exercise 3.3.4 A. Graph of a function \(F\) for Exercise 3.3.4a. B. Graph of a function \(G\) for Exercise 3.3.4b.

Exercise 3.3.5 Danger: Obnubilation Zone.

1. Shown in Exercise Figure 3.3.5A is a graph of the derivative, \(F ^{\prime}\) , of a function, \(F\). One point of the graph of \(F\) is (0,2) (that is, \(F(0) = 2\)). Your job, should you accept it, is to sketch a reasonable graph of \(F\).
2. Repeat the steps of a. to sketch a graph of a function \(G\) knowing that \(G(1) = 0\) and the graph of \(G ^{\prime} (x)\) is shown in Exercise Figure 3.3.5B. (Consider: What is the slope of \(G\) at \(x = 1\)? What is the slope of \(G\) at \(x = 2\)? What is the slope of \(G\) at \(x = 3\)?)

Figure for Exercise 3.3.5 A. Graph of \(F ^{\prime}\) for a function \(F\) for Exercise 3.3.5a. B. Graph of \(G ^{\prime}\) for a function \(G\) for Exercise 3.3.5b.

Exercise 3.3.6 In Figure 3.3.1A it appears that at \(t = 6\) years the population was about 8.75 thousand, or 8,750 individuals and that the growth rate was about 0.75 thousand per year, or 750 individuals per year. Suppose this is a deer population and you wished to allow hunters to harvest some deer each year. How many deer could be harvested each year and the population size remain at 8,750 individuals?

Exercise 3.3.7 Technology. The function

\[P(t)=\frac{10 \cdot 2^{t}}{9+2^{t}}=\frac{10}{\left(9 \cdot 2^{-t}+1\right)}\]

is an example of a logistic function, a type of function that often is used to describe the growth of populations. Plot the graph of this function for \(0 \leq t \leq 10, 0 \leq y \leq 20\). Find how to plot \(P ^{\prime} (t)\) and locate the highest point of \(P ^{\prime} (t)\) on your technology. MATLAB code is

Code \(\PageIndex{1}\) (MATLAB):

close all;clc;clear; h=0.001; x=[0:h:5]; y=10./(9*2.^(-x)+1);
z=diff(y)/h; [m,i]=max(z); [x(i) y(i) z(i)]
plot(x,y,’linewidth’,2); hold(’on’)
plot(x(i),y(i),’x’,’linewidth’,3)
plot(x(1:length(x)-1),z,’r’,’linewidth’,2)
plot(x(i),z(i),’xr’,’linewidth’,3)

At what time and population size is the population growing the fastest?

Exercise 3.3.8 The square function, \(S(t)=t^{2}\), \(t \geq 0\), and the square root function, \(R(t)=\sqrt{t}\), \(t \geq 0\), are each inverses of the other. See Figure Ex. 3.3.8. Compare the slopes of the tangents to \(S\) at the points (2,4), (3,9), and (4,16) with the slopes of \(R\) at the respectively corresponding points, (4,2), (9,3), and (16,4) of \(R\). Compare the slope of the graph of \(S\) at the point (\(a, a^{2}\)), \(a > 0\) with the slope of the graph of \(R\) at the corresponding point (\(a^{2}, a\)).

Figure for Exercise 3.3.8 Graphs of \(S(t)=t^{2}\) and \(R(t)=\sqrt{t}\). See Exercise 3.3.8.

The next two exercises explore the reflective property of a parabola which asserts that light rays originating from the focal point of a parabola will strike the parabola and be reflected in a direction parallel to the axis of the parabola. We choose the parabola that is the graph of \(y = 2\sqrt{t}\) which has (1,0) as its focal point and \(x = −1\) as its directrix.

Figure \(\PageIndex{3}\): A parabolic reflector used to boil water in the kettle that is at the focal point of the dish, in Lhasa, Tibet.

Exercise 3.3.9 Shown in Figure Ex. 3.3.9 is the graph of \(y = 2\sqrt{t}\) and a ray emanating vertically from the focal point at (1,0) and reflected (apparently horizontally) by the tangent to the parabola at (1,2). The angles A (of incidence) and B (of reflection) are equal.

1. Compute the slope of the tangent to \(y = 2\sqrt{t}\) at the point (1,2).
2. Argue that the angle A in Figure Ex. 3.3.9 is \(45^{\circ}\).
3. Argue that the angle B in Figure Ex. 3.3.9 is \(45^{\circ}\).
4. Argue that the reflected ray from (1,2) is horizontal.

Figure for Exercise 3.3.9 Graph of the parabola \(y = 2\sqrt{t}\) and a ray (dashed line) emanating vertically from the focal point (1,0) and reflected at (1,2). See Exercise 3.3.9.

Exercise 3.3.10 Shown in Figure Ex. 3.3.10 is the graph of \(y = 2\sqrt{t}\), and a ray (dashed line) from the focal point, (1,0), to the point \((a, 2 \sqrt{a})\), and a tangent, \(T\), to the parabola at \((a, 2 \sqrt{a})\). Our goal is to show that the reflected ray (dashed line with arrow head) is horizontal (but it has not been drawn that way). The two angles marked \(\beta\) are equal because they are vertical angles of intersecting lines.

It will be sufficient to show that the angle of reflection, \(B\), is also \(\beta\), the angle of inclination of the tangent \(T\). Because \(B\) and \(\beta\) are acute, it will be sufficient to show that \(\tan{B} = \tan{\beta}\).

1. Argue that \(C=A+\beta\), so that \(A = C − \beta\).
2. Argue that \(B = A\), so that \(B = C − \beta\).
3. Compute the slope of the tangent \(T\) to the graph of \(y = 2\sqrt{t}\) at \((a, 2 \sqrt{a})\). By definition, this number is also \(\tan{\beta}\).
4. Compute \(\tan{C}\).
5. Use the trigonometric identity \[\tan (C-\beta)=\frac{\tan C-\tan \beta}{1+\tan C \tan \beta}\] to show that \(\tan B=\tan (C-\beta)=\frac{1}{\sqrt{a}}\).

Because \(\tan B=\tan \beta, B=\beta\) and the reflected ray is horizontal.

Figure for Exercise 3.3.10 Graph of the parabola \(y = 2 \sqrt{t}\) and a ray (dashed line) emanating vertically from the focal point (1,0) and reflected at a point \((a, 2 \sqrt{a})\). See Exercise 3.3.10.

 

⁴ Joseph-Louis Lagrange (a French mathematician of Italian descent, 1736-1813) used the word ‘derivative’, and may have been the first to do so (H. L. Vacher, Computational geology 5 – If geology, then calculus, J. Geosci. Educ., 1999, 47 186-195).

⁵ Abraham Robinson, Nonstandard Analysis, Princeton University Press, 1996 is a recent book defining calculus based on infinitesimals.