<< Chapter < Page Chapter >> Page >
  • State the constant, constant multiple, and power rules.
  • Apply the sum and difference rules to combine derivatives.
  • Use the product rule for finding the derivative of a product of functions.
  • Use the quotient rule for finding the derivative of a quotient of functions.
  • Extend the power rule to functions with negative exponents.
  • Combine the differentiation rules to find the derivative of a polynomial or rational function.

Finding derivatives of functions by using the definition of the derivative can be a lengthy and, for certain functions, a rather challenging process. For example, previously we found that d d x ( x ) = 1 2 x by using a process that involved multiplying an expression by a conjugate prior to evaluating a limit. The process that we could use to evaluate d d x ( x 3 ) using the definition, while similar, is more complicated. In this section, we develop rules for finding derivatives that allow us to bypass this process. We begin with the basics.

The basic rules

The functions f ( x ) = c and g ( x ) = x n where n is a positive integer are the building blocks from which all polynomials and rational functions are constructed. To find derivatives of polynomials and rational functions efficiently without resorting to the limit definition of the derivative, we must first develop formulas for differentiating these basic functions.

The constant rule

We first apply the limit definition of the derivative to find the derivative of the constant function, f ( x ) = c . For this function, both f ( x ) = c and f ( x + h ) = c , so we obtain the following result:

f ( x ) = lim h 0 f ( x + h ) f ( x ) h = lim h 0 c c h = lim h 0 0 h = lim h 0 0 = 0.

The rule for differentiating constant functions is called the constant rule    . It states that the derivative of a constant function is zero; that is, since a constant function is a horizontal line, the slope, or the rate of change, of a constant function is 0 . We restate this rule in the following theorem.

The constant rule

Let c be a constant.

If f ( x ) = c , then f ( c ) = 0 .

Alternatively, we may express this rule as

d d x ( c ) = 0.

Applying the constant rule

Find the derivative of f ( x ) = 8 .

This is just a one-step application of the rule:

f ( 8 ) = 0.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the derivative of g ( x ) = −3 .

0

Got questions? Get instant answers now!

The power rule

We have shown that

d d x ( x 2 ) = 2 x and d d x ( x 1 / 2 ) = 1 2 x 1 / 2 .

At this point, you might see a pattern beginning to develop for derivatives of the form d d x ( x n ) . We continue our examination of derivative formulas by differentiating power functions of the form f ( x ) = x n where n is a positive integer. We develop formulas for derivatives of this type of function in stages, beginning with positive integer powers. Before stating and proving the general rule for derivatives of functions of this form, we take a look at a specific case, d d x ( x 3 ) . As we go through this derivation, pay special attention to the portion of the expression in boldface, as the technique used in this case is essentially the same as the technique used to prove the general case.

Differentiating x 3

Find d d x ( x 3 ) .

d d x ( x 3 ) = lim h 0 ( x + h ) 3 x 3 h = lim h 0 x 3 + 3 x 2 h + 3 x h 2 + h 3 x 3 h Notice that the first term in the expansion of ( x + h ) 3 is x 3 and the second term is 3 x 2 h . All other terms contain powers of h that are two or greater. = lim h 0 3 x 2 h + 3 x h 2 + h 3 h In this step the x 3 terms have been cancelled, leaving only terms containing h . = lim h 0 h ( 3 x 2 + 3 x h + h 2 ) h Factor out the common factor of h . = lim h 0 ( 3 x 2 + 3 x h + h 2 ) After cancelling the common factor of h , the only term not containing h is 3 x 2 . = 3 x 2 Let h go to 0.
Got questions? Get instant answers now!
Got questions? Get instant answers now!
Practice Key Terms 7

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask