3 differentiation rules. we have: seen how to interpret derivatives as slopes and rates of change ...

3DIFFERENTIATION RULESDIFFERENTIATION RULES

We have:

Seen how to interpret derivatives as slopes and rates of change

Seen how to estimate derivatives of functions given by tables of values

Learned how to graph derivatives of functions that are defined graphically

Used the definition of a derivative to calculate the derivatives of functions defined by formulas

DIFFERENTIATION RULES

These differentiation rules enable

us to calculate with relative ease

the derivatives of: Polynomials

Rational functions

Algebraic functions

Exponential and logarithmic functions

Trigonometric and inverse trigonometric functions

DIFFERENTIATION RULES

3.1Derivatives of Polynomials

and Exponential Functions

In this section, we will learn:

How to differentiate constant functions, power functions,

polynomials, and exponential functions.

DIFFERENTIATION RULES

Let’s start with the simplest of

all functions—the constant function

f(x) = c.

CONSTANT FUNCTION

The graph of this function is the

horizontal line y = c, which has slope 0.

So, we must have f’(x) = 0.

CONSTANT FUNCTION

A formal proof—from the definition of

a derivative—is also easy.

0

0 0

( )'( ) lim

lim lim0 0

h

h h

f x h f xf x

hc c

h

CONSTANT FUNCTION

We next look at the functions

f(x) = xn, where n is a positive

integer.

POWER FUNCTIONS

If n = 1, the graph of f(x) = x is the line

y = x, which has slope 1.

So,

You can also verify Equation 1 from the definition of a derivative.

( ) 1dx

dx

POWER FUNCTIONS Equation 1

We have already investigated the cases

n = 2 and n = 3.

In fact, in Section 2.8, we found that:

2 3 2( ) 2 ( ) 3d dx x x x

dx dx

POWER FUNCTIONS Equation 2

For n = 4 we find the derivative of f(x) = x4

as follows:

4 4

0 0

4 3 2 2 3 4 4

0

3 2 2 3 4

0

3 2 2 3 3

0

( ) ( ) ( )'( ) lim lim

4 6 4lim

4 6 4lim

lim 4 6 4 4

h h

h

h

h

f x h f x x h xf x

h h

x x h x h xh h x

h

x h x h xh h

h

x x h xh h x

POWER FUNCTIONS

If n is a positive integer,

then1( )n nd

x nxdx

POWER RULE

In finding the derivative of x4, we had to

expand (x + h)4.

Here, we need to expand (x + h)n .

To do so, we use the Binomial Theorem—as follows.

0 0

( ) ( ) ( )'( ) lim lim

n n

h h

f x h f x x h xf x

h h

Proof 2POWER RULE

This is because every term except the first has h as a factor and therefore approaches 0.

1 2 2 1

0

1 2 2 1

0

1 2 2 1 1

0

( 1)2

'( ) lim

( 1)2lim

( 1)lim

2

n n n n n n

h

n n n n

h

n n n n n

h

n nx nx h x h nxh h x

f xh

n nnx h x h nxh h

hn n

nx x h nxh h nx

POWER RULE Proof 2

a.If f(x) = x6, then f’(x) = 6x5

b.If y = x1000, then y’ = 1000x999

c.If y = t4, then

d. = 3r23( )

dr

dr

Example 1POWER RULE

34dy

tdt

What about power functions with

negative integer exponents?

In Exercise 61, we ask you to verify from the definition of a derivative that:

We can rewrite this equation as:

2

1 1d

dx x x

1 21dx x

dx

NEGATIVE INTEGERS

What if the exponent is a fraction?

In Example 3 in Section 2.8, we found that:

This can be written as:

1

2

dx

dx x

1 2 1 212

dx x

dx

FRACTIONS

If n is any real number,

then1( )n nd

x nxdx

POWER RULE—GENERAL VERSION

Differentiate:

a.

b.

In each case, we rewrite the function as a power of x.

2

1( )f x

x

3 2y x

Example 2POWER RULE

Since f(x) = x-2, we use the Power Rule

with n = -2:

Example 2 a

2

2 1

33

'( ) ( )

2

22

df x x

dx

x

xx

POWER RULE

3 2

2 3

2 3 1 1 323

( )

( )

2

3

dy dx

dx dxdx

dx

x x

POWER RULE Example 2 b

It also enables us to find normal

lines.

The normal line to a curve C at a point P is the line through P that is perpendicular to the tangent line at P.

In the study of optics, one needs to consider the angle between a light ray and the normal line to a lens.

NORMAL LINES

Find equations of the tangent line and normal

line to the curve at the point (1, 1).

Illustrate by graphing the curve and these

lines.

Example 3TANGENT AND NORMAL LINES

y x x

The derivative of is:

So, the slope of the tangent line at (1, 1) is:

Thus, an equation of the tangent line is:

or

321 2( )f x x x xx x

3 2 1 1/ 23 3 32 2 2'( )f x x x x

32'(1)f

321 ( 1)y x 3 1

2 2y x

Example 3TANGENT LINE

The normal line is perpendicular to

the tangent line.

So, its slope is the negative reciprocal of , that is .

Thus, an equation of the normal line is:

or231 ( 1)y x 52

3 3y x

Example 3NORMAL LINE

322

3

We graph the curve and its tangent line

and normal line here.

Example 3TANGENT AND NORMAL LINES

If c is a constant and f is a differentiable

function, then

( ) ( )d dcf x c f x

dx dx

CONSTANT MULTIPLE RULE

Let g(x) = cf(x).

Then,0

0

0

0

( ) ( )'( ) lim

( ) ( )lim

( ) ( )lim

( ) ( )lim (Law 3 of limits)

'( )

h

h

h

h

g x h g xg x

hcf x h cf x

hf x h f x

ch

f x h f xc

hcf x

ProofCONSTANT MULTIPLE RULE

4 4

3 3

.a. (3 ) 3 ( )

3(4 ) 12

b. ( ) ( 1)

1 ( ) 1(1) 1

d dx x

dx dx

x x

d dx x

dx dxdx

dx

NEW DERIVATIVES FROM OLD Example 4

If f and g are both differentiable,

then

( ) ( ) ( ) ( )d d df x g x f x g x

dx dx dx

SUM RULE

Let F(x) = f(x) + g(x). Then,

0

0

0

0 0

( ) ( )'( ) lim

( ) ( ) ( ) ( )lim

( ) ( ) ( ) ( )lim

( ) ( ) ( ) ( )lim lim (Law 1)

'( ) '( )

h

h

h

h h

F x h F xF x

hf x h g x h f x g x

hf x h f x g x h g x

h h

f x h f x g x h g x

h hf x g x

ProofSUM RULE

If f and g are both differentiable,

then

( ) ( ) ( ) ( )d d df x g x f x g x

dx dx dx

DIFFERENCE RULE

8 5 4 3

8 5 4

3

7 4 3 2

7 4 3 2

( 12 4 10 6 5)

12 4

10 6 5

8 12 5 4 4 10 3 6 1 0

8 60 16 30 6

dx x x x x

dxd d dx x x

dx dx dxd d dx x

dx dx dx

x x x x

x x x x

NEW DERIVATIVES FROM OLD Example 5

Find the points on the curve

y = x4 - 6x2 + 4

where the tangent line is horizontal.

NEW DERIVATIVES FROM OLD Example 6

Horizontal tangents occur where

the derivative is zero.

We have:

Thus, dy/dx = 0 if x = 0 or x2 – 3 = 0, that is, x = ± .

4 2

3 2

( ) 6 ( ) (4)

4 12 0 4 ( 3)

dy d d dx x

dx dx dx dx

x x x x

NEW DERIVATIVES FROM OLD Example 6

3

So, the given curve has horizontal tangents

when x = 0, , and - .

The corresponding points are (0, 4), ( , -5), and (- , -5).

NEW DERIVATIVES FROM OLD Example 6

33

3 3

The equation of motion of a particle is

s = 2t3 - 5t2 + 3t + 4, where s is measured

in centimeters and t in seconds.

Find the acceleration as a function of time.

What is the acceleration after 2 seconds?

NEW DERIVATIVES FROM OLD Example 7

The velocity and acceleration are:

The acceleration after 2 seconds is: a(2) = 14 cm/s2

2( ) 6 10 3

( ) 12 10

dsv t t t

dtdv

a t tdt

NEW DERIVATIVES FROM OLD Example 7

Let’s try to compute the derivative of

the exponential function f(x) = ax using

the definition of a derivative:

0 0

0 0

( ) ( )'( ) lim lim

( 1)lim lim

x h x

h h

x h x x h

h h

f x h f x a af x

h h

a a a a a

h h

EXPONENTIAL FUNCTIONS

The factor ax doesn’t depend on h.

So, we can take it in front of the limit:

0

1'( ) lim

hx

h

af x a

h

EXPONENTIAL FUNCTIONS

41

Calculating the derivative of g(x) = 2x

0

2 1lim ?

h

hBut what is

h

(2 )0.693 2 2

xx xdk

dx

: ln 2 .6931Note

42

Similar calculations show the following for other values of a:

( )x xd ak a

dx

: ln 2 .6931Note

ln 3 1.0986ln 4 1.3863ln 5 1.6094ln 6 1.7918ln 7 1.9459

Thus, we have:

(2 ) ln 2 2 (3 ) ln 3 3x x x xd d

dx dx

EXPONENTIAL FUNCTIONS Equation 5

Consider 0

1lim ,

h

h

e

h

DEFINITION OF THE NUMBER e

1

.001 .9995

.0001 .99995

.00001 .999995

.00001 1.000005

.0001 1.00005

.001 1.0005

heh

h

So it appears that .

0

1lim 1

h

h

e

h

Geometrically, this means that, of all the

possible exponential functions y = ax, the

function f(x) = ex is the one whose tangent line

at (0, 1) has a slope f’(0) that is exactly 1.

EXPONENTIAL FUNCTIONS

The derivative of the natural exponential

function is:

( )

( ) (ln )

x x

x x

de e

dx

da a a

dx

DERIV. OF NATURAL EXPONENTIAL FUNCTION

Thus, the exponential function f(x) = ex

has the property that it is its own derivative.

The geometrical significance of this fact is that the slope of a tangent line to the curve y = ex is equal to the y coordinate of the point.

EXPONENTIAL FUNCTIONS

If f(x) = ex - x, find f’ .

Compare the graphs of f and f’.

EXPONENTIAL FUNCTIONS Example 8

'( ) ( )

( ) ( )

1

x

x

x

df x e x

dxd de x

dx dx

e

The function f and its derivative f’ are

graphed here.

Notice that f has a horizontal tangent when x = 0.

This corresponds to the fact that f’(0) = 0.

EXPONENTIAL FUNCTIONS Example 8

Notice also that, for x > 0, f’(x) is positive

and f is increasing.

When x < 0, f’(x) is negative and f

is decreasing.

EXPONENTIAL FUNCTIONS Example 8