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Appendix to Chapter 2 Calculus and Optimization Techniques 65

independent variable. We write y f(x), where f(x) represents the (unspeci-

fied) functional relationship between the variables. The notation dy/dx rep-

resents the derivative of the function, that is, the rate of change or slope of the

function at a particular value of x. (The d in this notation is derived from the

Greek letter delta, which has come to mean “change in.”) We list the following

basic rules.

Rule 1.The derivative of a constant is zero. If y 7, for example,

dy/dx 0. Note that y 7 is graphed as a horizontal line (of height 7);

naturally this has a zero slope for all values of x.

Rule 2.The derivative of a constant times a variable is simply the

constant. If y bx, then dy/dx b. For example, if y 13x, then

dy/dx 13. In words, the function y 13x is a straight line with a

slope of 13.

Rule 3.A power function has the form y axn, where a and n are

constants. The derivative of a power function is

For instance, if y 4x^3 , then dy/dx 12x^2.

It is important to recognize that the power function includes many

important special cases.^2 For instance, is equivalently written as

. Similarly, becomes. According to Rule 3, the respective
derivatives are and
Rule 4.The derivative of a sum of functions is equal to the sum of the
derivatives; that is, if y f(x) g(x), then dy/dx df/dx dg/dx.
This simply means we can take the derivative of functions term by
term. For example, given that y .1x^2 2x^3 , then dy/dx .2x 6x^2.
Rule 5.Suppose y is the product of two functions: y f(x)g(x).
Then we have

For example, suppose we have y (4x)(3x^2 ). Then dy/dx (4)(3x^2 ) 

4x(6x) 36x^2. (Note that this example can also be written as y 12x^3 ; we

confirm that dy/dx 36x^2 using Rule 3.)

Rule 6.Suppose y is a quotient: y f(x)/g(x). Then we have

dy/dx 3 (df/dx)g(dg/dx)f 4 /g^2.

dy/dx(df/dx)g(dg/dx)f.

dy/dx2x^3 2/x^3 dy/dx.5x1/2.5/ 1 x.

yx^2 y 1 x yx 1/2

y1/x^2

dy/dxn#axn^1.

(^2) Notice that Rules 1 and 2 are actually special cases of Rule 3. Setting n 0 implies that y a, and,
therefore, dy/dx 0 (Rule 1). Setting n 1 implies that y ax and, therefore, dy/dx a (Rule 2).
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