3.3 Rules for Differentiation

Positive Integer Powers, Multiples, Sums and DifferencesProducts and QuotientsNegative Integer Powers of xSecond and Higher Order Derivatives

Derivative of a Constant

The notation in your book might look different from what you learned before, but you should be able to switch back and forth between notations.

Positive Integer Powers, Multiples

3 2

Example:

3dx x

dx

3 2 2

Example:

5 5 3 15d

x x xdx

2

2 2

Example:

Let 3 and 2

3 2 3 2

6 2

u x v x

d d dx x x x

dx dx dxx

Sums and differences

The Product Rule

( ) '( ) ( ) '( )u x v x v x u x

Try these:

2 31. Find ' if ( ) 1 3f f x x x

2.

The Quotient Rule

2

( ) '( ) ( ) '( )

( )

v x u x u x v x

v x

Try this:2

2

1Find ' if ( ) and support your answer graphically.

1

xf f x

x

Negative integer powers of x

2

Example:

3Let . Find by

a. using the quotient rule:

b. dividing the terms in the numerator by the denominator and then differentiating.

x dyy

x dx

Higher order derivativesFor the second derivative of y:

When finding the “nth” derivative, we use the notation:

Example:

Find the 55th derivative of cos .y x

Example 9: Finding Instantaneous Rates of Change An orange farmer currently has 200 trees yielding an average of 15

bushels of oranges per tree. She is expanding her farm at the rate of 15 trees per year, while improved husbandry is improving her average annual yield by 1.2 bushels per tree. What is the current (instantaneous) rate of increase of her total annual production of oranges?

The function f is differentiable for all real numbers. Estimate the following:

x 0 2 4 6 8

f(x) 9 6 5 3 4

(5) ______f

4 ________x

df

dx

8 ________x

df

dx

Derivatives from Numerical Data:

( )c. Let ( ) . Find '(2).

( )

f xk x k

g x

1.1h

Assignment

3.3A: p.124:1,5,13,17,21,29,32,52,56 and p. 126: Quick Quiz: 1,2

3.3B: p. 124: 7,23,27,38,41,46,49,51; Read Section 3.4