3.3 Differentiation Formulas

In this section we'll use the definition of the derivative to derive some rules for finding derivatives

Derivative of a Constant Function

Prove using    f '(x) =

Power Functions

Earlier we have found that

Find

Find

Let's use the definition of the derivative to find

The Power Rule         If n is a positive integer,

There are two versions of the proof in the text - we'll do the binomial

Recall the binomial expansion:

Using    f '(x) =

Examples:

Constant Multiple Rule

If c is a constant and f is a differentiable function, then

Prove:  Let g(x) = cf(x) and use Definition of derivative

Examples:

The Sum Rule   If f and g are both differentiable, then

Prove:  Let m(x) = f(x) + g(x)

The Difference Rule   If f and g are both differentiable, then

Proof is similar to The Sum Rule.

Example:

The Product Rule

If f and g are both differentiable, then

Alternately:  f ' g + g' f

Proof:  This one gets Interesting!!    Let m(x) = f(x)g(x)

Clever device used:  Subtract and add a term

Example:

If f(3)= 4 and f ’(3) = -2, g(3) = 7  and g’(3) = 5,  find

Quotient Rule                   Alternately:

If f and g are differentiable, then

Again, the proof requires add and subtract a term  -

Example:

The Expanded Power Function Rule

If n is a positive integer, then

For proof, recall that , and use the quotient rule from above.

Example:

The previous version of the power rule was only for integers, but we did find that

The Power Rule (General Version)

If n is any real number, then

Proof is done in Chapter 7.

Examples:

Find f’(x):

Assignment:  3.3  pg. 144, 1-23 odd, 27-31 odd, 35-41 odd, 49, 55-63 odd, 69, 71