AP CalculusChapter 3 Section 3RULES FOR DIFFERENTIATION

Rule 1: Derivative of a Constant The first rule of differentiation is that the derivative of every constant function is the zero of the function.

If f(x) = c, then

Proof:

0 cdxd

dxdf

00limlim)()(lim000

hhh hcc

hxfhxf

Rule 2: Power Rule for Positive Integer Powers of x If n is a positive integer, then

The proof is on page 116.

1)( nn nxxdxd

Rule 2: Examples

34

23

12

4

3

2

xxdxd

xxdxd

xxdxd

Rule 3: Constant Multiple Rule If u is a differentiable function of x and c is a constant, then

Example:

dxduccu

dxd

)(

3344 284777 xxxdxdx

dxd

Rule 4: Sum & Difference Rule If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points,

Example:

This is a very powerful rule because it allows us to find the derivatives of polynomials.

dxdv

dxduvu

dxd

xxdxdx

dxdx

dxdxx

dxd 442222 32424

Rule 5: Product Rule The product of 2 differentiable functions u and v is differentiable, and

“the first times the derivative of the second plus the second times the derivative of the first”

Note: the order you do the differentiation does not matter because we are adding the two together.

dxduv

dxdvuuv

dxd

Product Rule Example Find the derivative of Solution: Let

Rule 6: Quotient Rule At a point where v ≠ 0, the quotient y = u/v of 2 differentiable functions is differentiable, and

“the bottom times the derivative of the top minus the top times the derivative of the bottom, all over the bottom squared”

2vdxdvu

dxduv

vu

dxd

Quotient Rule Example Differentiate Solution: Let

Rule 7: Power Rule for Negative Integer Powers of x If n is a negative integer and x ≠ 0, then

We will use this rule after we discuss the Chain Rule. This rule can be used to solve for a derivative in place of the Quotient Rule because we can redefine the denominator as a negative integer power of x.

1 nn nxxdxd

Second & Higher Order Derivatives We can find the derivative of the derivative, which is called the second derivative. Denoted:

We can also find the third, fourth, fifth, and nth derivatives. Denoted:

2

2

dxyd

dxdy

dxd

dxydy

n

nnn

dxydy

dxdy

dxyd

dxydy

)1()(

3

3

Example Find the first four derivatives of y = x3 – 5x2 + 2

Example Find the first four derivatives of y = x3 – 5x2 + 2

0

6106103

)4(

2

y

yxy

xxy

Finding Instantaneous Rate 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?

Solution Let the functions t and y be defined as follows:

Then, We know: We need to find Apply Product Rule:

Homework # 3 – 52 by multiples of 3 on pages 124 - 125