# Derivative

# Derivative

In mathematics, the derivative is the exact rate at which one quantity changes with respect to another. The Greek symbol delta (Δ) is usually used to represent

Table 1. Derivatives . (Thomson Gale.) | ||||||
---|---|---|---|---|---|---|

Approximation (using derivatives) of line segments | ||||||

_{x}1 | _{x}2 | _{t}1 | _{t}2 | _{x}2- _{x}1 | _{t}2- _{t}1 | (_{x}2- _{x}1 )/(_{t}2- _{t}1 ) |

0 | 8 | 0 | 0.707106781 | 8 | 0.707106781 | 11.3137085 |

1 | 7 | 0.25 | 0.661437828 | 6 | 0.411437828 | 14.58300524 |

3 | 5 | 0.433012702 | 0.559016994 | 2 | 0.126004292 | 15.87247514 |

the change when using the derivative. Geometrically, the derivative is the slope of a curve at a point on the curve, defined as the slope of the tangent to the curve at the same point. The process of finding the derivative is called differentiation. This process is central to the branch of mathematics called differential calculus. It is also one of the two key concepts of calculus, the other being the integral. Integral calculus (or integration) and differential calculus are based on the fundamental theorem of calculus.

## History and usefulness

Calculus was developed independently by English physicist and mathematician Sir Isaac Newton (1642–1727) and German mathematician Gottfried Wilhelm Leibniz (1646–1716) around the middle part of the seventeenth century. Newton was a physicist as well as a mathematician. He found that the mathematics of his time was not sufficient to solve the problems he was interested in, so he invented new mathematics. About the same time, another mathematician, Leibniz, developed the same ideas as Newton. Newton was interested in calculating the velocity of an object at any instant. For example, if a person sits under an apple tree, as legend has it Newton did, and an apple falls and hits the person’s head, that person might ask how fast the apple was traveling just before impact. More importantly, many of today’s scientists are interested in calculating the rate at which a satellite’s position changes with respect to time (its rate of speed). Most investors are interested in how a stock’s value changes with time (its rate of growth). In fact, many of today’s important problems in the fields of physics, chemistry, engineering, economics, biology, and the other sciences involve finding the rate at which one quantity changes with respect to another, that is, they involve finding the derivative.

## The basic concept

The derivative is often called the instantaneous rate of change. A rate of change is simply a comparison of the change in one quantity to the simultaneous change in a second quantity. For instance, the amount of money an employer owes an employee compared to the length of time the worker worked for the money determines the rate of pay. The comparison is made in the form of a ratio, dividing the change in the first quantity by the change in the second quantity. When both changes occur during an infinitely short period of time (in the same instant), the rate is said to be instantaneous, and then the ratio is called the derivative.

To better understand what is meant by an instantaneous rate of change, consider the graph of a straight line (Figure 1).

The line’s slope is defined to be the ratio of the rise (vertical change between any two points) to the run (simultaneous horizontal change between the same two points). This means that the slope of a straight line is a rate, specifically, the line’s rate of rise with respect to the horizontal axis. It is the simplest type of rate because it is constant, the same between any two points, even two points that are arbitrarily close together. Roughly speaking, arbitrarily close together means one can make them closer than any positive amount of separation. The derivative of a straight line, then, is the same for every point on the line and is equal to the slope of the line.

Determining the derivative of a curve is somewhat more difficult, because its instantaneous rate of rise changes from point to point (Figure 2).

One can estimate a curve’s rate of rise at any particular point, though, by noticing that any section of a curve can be approximated by replacing it with a straight line. Since one knows how to determine the slope of a straight line, an approximation of a curve’s rate of rise at any point can be made by determining the slope of an approximating line segment. The shorter the approximating line segment becomes, the more accurate the estimate becomes. As the length of the approximating line segment becomes arbitrarily short, so does its rise and its run. Just as in the case of

the straight line, an arbitrarily short rise and run can be shorter than any given positive pair of distances. Thus, their ratio is the instantaneous rate of rise of the curve at the point or the derivative. In this case, the derivative is different at every point, and equal to the slope of the tangent at each point. (A tangent is a straight line that intersects a curve at a single point.)

## A concrete example

A fairly simple, and not altogether impractical example is that of the falling apple. Observation tells one that the apple’s initial speed (the instant before letting go from the tree) is zero, and that it accelerates rapidly. Scientists have found, from repeated measurements with various falling objects (neglecting wind resistance), that the distance an object falls toward the Earth (call it S) in a specified time period (call it T) is given by the following equation (Figure 2):

(1) S = 16 T^{2}

Suppose one is interested in the apple’s speed after it has dropped 4 ft (1.2 m). As a first approximation, connect the points where Sl_{1} = 0 and Sl_{2} = 8 (Figure 3 and line 1 of Table 1).

Using equation (1), find the corresponding times, and calculate the slope of the approximating line segment (use the formula in Figure 1). Repeat this process numerous times, each time letting the two points get closer together. If a calculator or computer spreadsheet is available this is rather simple. Table 1 shows the result for several approximating line segments. The line segments corresponding to the first two entries in the table are drawn in Figure 3. Looking at Figure 3, it is clear that as the approximating line gets shorter, its slope approximates the rate of rise of the curve more accurately.

### KEY TERMS

**Infinitesimal—** Approaching zero in length. When the separation between two points approaches zero but never quite gets there, the separation is said to be infinitesimal.

**Instantaneous—** Occurring in an instant or an infinitely short period of time.

**Rate—** A rate is a comparison of the change in one quantity with the simultaneous change in another, where the comparison is made in the form of a ratio.

**Ratio—** The fraction formed when two quantities are compared by division, that is, one quantity is divided by the other.

**Slope—** Slope is the ratio of the vertical distance separating any two points on a line, to the horizontal distance separating the same two points.

## Resources

### BOOKS

Burton, David M. *The History of Mathematics: An Introduction*. New York: McGraw-Hill, 2007.

Downing, Douglas. *Calculus the Easy Way*. 4th ed. Hauppauge, New York: Barron’s Educational Services, Inc., 2006.

Dunham, William. *The Calculus Gallery: Masterpieces from Newton to Lebesgue*. Princeton, NJ: Princeton University Press, 2005.

J. R. Maddocks

# Derivative

# Derivative

In **mathematics** , the derivative is the exact **rate** at which one quantity changes with respect to another. Geometrically, the derivative is the slope of a **curve** at a **point** on the curve, defined as the slope of the tangent to the curve at the same point. The process of finding the derivative is called differentiation. This process is central to the branch of mathematics called differential **calculus** .

## History and usefulness

Calculus was first invented by Sir Isaac Newton around 1665. Newton was a physicist as well as a mathematician. He found that the mathematics of his time was not sufficient to solve the problems he was interested in, so he invented new mathematics. About the same time another mathematician, Goltfried Leibnez, developed the same ideas as Newton. Newton was interested in calculating the **velocity** of an object at any instant. For example, if you sit under an apple **tree** , as legend has it Newton did, and an apple falls and hits you on the head, you might ask how fast the apple was traveling just before impact. More importantly, many of today's scientists are interested in calculating the rate at which a satellite's position changes with respect to time (its rate of speed). Most investors are interested in how a stock's value changes with time (its rate of growth). In fact, many of today's important problems in the fields of **physics** , **chemistry** , **engineering** , economics, and **biology** involve finding the rate at which one quantity changes with respect to another, that is, they involve finding the derivative.

## The basic concept

The derivative is often called the "instantaneous" rate of change. A rate of change is simply a comparison of the change in one quantity to the simultaneous change in a second quantity. For instance, the amount of money your employer owes you compared to the length of time you worked for him determines your rate of pay. The comparison is made in the form of a **ratio** , dividing the change in the first quantity by the change in the second quantity. When both changes occur during an infinitely short period of time (in the same instant), the rate is said to be "instantaneous," and then the ratio is called the derivative.

To better understand what is meant by an instantaneous rate of change, consider the graph of a straight line (see Figure 1).

The line's slope is defined to be the ratio of the rise (vertical change between any two points) to the run (simultaneous horizontal change between the same two points). This means that the slope of a straight line is a rate, specifically, the line's rate of rise with respect to the horizontal axis. It is the simplest type of rate because it is constant, the same between any two points, even two points that are arbitrarily close together. Roughly speaking, arbitrarily close together means you can make them closer than any positive amount of separation. The derivative of a straight line, then, is the same for every point on the line and is equal to the slope of the line.

_{x}1 | _{x}2 | _{t}1 | _{t}2 | _{x}x2- 1 | _{t}t2- 1 | x( x2- t1)/( t2- 1) |

0 | 8 | 0 | 0.707106781 | 8 | 0.707106781 | 11.3137085 |

1 | 7 | 0.25 | 0.661437828 | 6 | 0.411437828 | 14.58300524 |

3 | 5 | 0.433012702 | 0.559016994 | 2 | 0.126004292 | 15.87247514 |

Determining the derivative of a curve is somewhat more difficult, because its instantaneous rate of rise changes from point to point (see Figure 2).

We can estimate a curve's rate of rise at any particular point, though, by noticing that any section of a curve can be approximated by replacing it with a straight line. Since we know how to determine the slope of a straight line, we can approximate a curve's rate of rise at any point, by determining the slope of an approximating line segment. The shorter the approximating line segment becomes, the more accurate the estimate becomes. As the length of the approximating line segment becomes arbitrarily short, so does its rise and its run. Just as in the case of the straight line, an arbitrarily short rise and run can be shorter than any given positive pair of distances. Thus, their ratio is the instantaneous rate of rise of the curve at the point or the derivative. In this case the derivative is different at every point, and equal to the slope of the tangent at each point. (A tangent is a straight line that intersects a curve at a single point.)

## A concrete example

A fairly simple, and not altogether impractical example is that of the falling apple. Observation tells us that the apple's initial speed (the instant before letting go from the tree) is **zero** , and that it accelerates rapidly. Scientists have found, from repeated measurements with
various falling objects (neglecting **wind** resistance), that the **distance** an object falls on the **earth** (call it S) in a specified time period (call it T) is given by the following equation (see Figure 2):

Suppose you are interested in the apple's speed after it has dropped 4 ft (1.2 m). As a first **approximation** , connect the points where Sl1=0 and Sl2=8 (see Figure 3 and line 1 of Table 1).

Using equation (1), find the corresponding times, and calculate the slope of the approximating line segment (use the formula in Figure 1). Repeat this process numerous times, each time letting the two points get closer together. If a **calculator** or computer spreadsheet is available this is rather simple. Table 1 shows the result for several approximating line segments.

The line segments corresponding to the first two entries in the table are drawn in Figure 3. Looking at Figure 3, it is clear that as the approximating line gets shorter, its slope approximates the rate of rise of the curve more accurately.

## Resources

### books

Allen, G.D., C. Chui, and B. Perry. *Elements of Calculus.* 2nd ed. Pacific Grove, CA: Brooks/Cole Publishing Co, 1989.

Boyer, Carl B. *A History of Mathematics.* 2nd ed. Revised by Uta C. Merzbach. New York: John Wiley and Sons, 1991.

Downing, Douglas. *Calculus the Easy Way.* 2nd ed. Hauppauge, NY: Barron's Educational Services, Inc., 1988.

### periodicals

McLaughlin, William I. "Resolving Zeno's Paradoxes." *Scientific American* 271 (1994): 84-89.

J. R. Maddocks

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Infinitesimal**—Approaching zero in length. When the separation between two points approaches zero but never quite gets there, the separation is said to be infinitesimal.

**Instantaneous**—Occurring in an instant or an infinitely short period of time.

**Rate**—A rate is a comparison of the change in one quantity with the simultaneous change in another, where the comparison is made in the form of a ratio.

**Ratio**—The fraction formed when two quantities are compared by division, that is, one quantity is divided by the other.

**Slope**—Slope is the ratio of the vertical distance separating any two points on a line, to the horizontal distance separating the same two points.

# derivative

de·riv·a·tive / diˈrivətiv/ •
adj. (typically of an artist or work of art) imitative of the work of another person, and usually disapproved of for that reason: *an artist who is not in the slightest bit derivative.* ∎ originating from, based on, or influenced by: *Darwin's work is derivative of the moral philosophers.* ∎ (of a financial product) having a value deriving from an underlying variable asset:

*equity-based derivative products.*• n. something that is based on another source:

*a derivative of the system was chosen for the Marine Corps’ V-22 tilt rotor aircraft.*∎ (often derivatives) an arrangement or instrument (such as a future, option, or warrant) whose value derives from and is dependent on the value of an underlying asset: [as adj.]

*the derivatives market.*∎ a word derived from another or from a root in the same or another language. ∎ a substance that is derived chemically from a specified compound:

*crack is a highly addictive cocaine derivative.*∎ Math. an expression representing the rate of change of a function with respect to an independent variable.DERIVATIVES: de·riv·a·tive·ly adv.

# derivative

**derivative** Rate of change of the value of a mathematical function with respect to a change in the independent variable. The derivative is an expression of the instantaneous rate of change of the function's value: in general it is itself a function of the variable. An example is obtaining the velocity and acceleration of an object that moves distance *x* in time *t* according to the equation x = at^{n}. In such motion, the velocity increases with time. The expression dx/dt, called the first derivative of distance with respect to time, is equal to the velocity of the object; in this example it equals nat^{(n−1)}. The result is obtained by differential calculus. In this example, the second derivative, written d^{2}x/dt^{2}, is equal to the acceleration.

# derivative

**derivative** of a formal language. The *left-derivative* of a language *L*, with respect to a word *w*, is {*w*′ | *ww*′ ∈ *L*}

where *ww*′ is the concatenation of *w* and *w*′. Similarly a *right-derivative* is {*w*′ | *w*′*w *∈ *L*}

# DERIVATIVE

**DERIVATIVE.** **1.** A WORD or other item of language that has been created according to a set of rules from a simpler word or item.

**2.** A COMPLEX WORD: *girlhood* from *girl*, *legal* from *leg-* (law), *legalize* from *legal*.

**3.** Of an essay, article, thesis, etc., and usually pejorative: depending for form and/or inspiration on an earlier and better piece of work.

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