In this section, we will consider the derivative function rather than just at a point. We also begin looking at some of the basic derivative rules.

Section 2.2 Derivatives of Power Functions and Polynomials

Definition (formal)

Let f be any function. The derivative function of f is defined as:

provided the limit exists

Other notations:

Example 1

Use the definition of derivative to find the derivative of the function .

Example 2

Use the definition of derivative to find the derivative of the function .

Example 3

Use the definition of derivative to find the derivative of the function .

Example 4

Use the definition of derivative to find the derivative of the function .

Theorem: Power Rule for Derivatives

Let n be any real number (not necessarily an integer).

Then:

Example 5

Find the derivative of each of the following functions.

(a)

(b)

(c)

Theorem: Constant Multiple Rule for

Derivatives

Let f be any differentiable function, let k be any constant, and let .

Then:

Theorem: Sum Rule for Derivatives

Let f and g be any differentiable functions, and let

. Then:

Example 6

Find the derivative of each of the following polynomials.

(a)

(b)

(c)

Example 7

Give the equation of the tangent line to the curve at the point (1, 3).