Section 2.2 – Basic Differentiation Rules and Rates of Change

The Constant RuleA constant function has derivative , or:

Note: The constant function is a horizontal line with a constant slope of 0.

f ' x 0

f x k

0ddx k

ExamplesDifferentiate both of the following functions.

f ' x ddx 13 0

a. f x 13

b. g x ei

The function is a horizontal line at y = 13. Thus the

slope is always 0.

The function is a horizontal line because e, Pi, and i all represent

numeric values. Thus the slope is always 0.

g' x ddx e

i 0

The Power RuleFor any real number n, the power function has the

derivative , or:

Ex:

f x x n

f ' x nx n 1

1n nddx x nx

f ' x 3x 2

f x x 3

ExamplesDifferentiate all of the following functions.

a. f x x 8

b. g u um1. Bring down the exponent

f ' x ddx x

8

8

x2. Leave the base alone 3. Subtract one from the original exponent.

8 1

8x 7

g' u ddu u

m

u

m 1

mThe procedure does not change with variables.

c. h x x3

x 2

Make sure the function is written as a power function to use the rule.

h' x ddx x

5 3

53

x

5 3 1

53 x

8 3

x1 3

x 2

x 5 3

The Constant Multiple RuleThe derivative of a constant times a function, is the constant

times the derivative of the function. In other words, if c is a constant and f is a differentiable function, then

Objective: Isolate a power function in order to take the derivative. For now, the cf(x) will look like .

ddx cf x c d

dx f x

cx n

ExamplesDifferentiate all of the following functions.

a. f x 3x 4

b. g x x“Pull out” the coefficient

f ' x ddx 3x 4

3

x

4 1

12x 3

1' 1ddxg x x

c. h x x 5

Make sure the function is written as a power function to use the rule.

h' x ddx x 5

x

5 1

5x 6

x 5

3 ddx x

4

4Take the derivative

1x1

11 ddx x

1

x

1 1

1

1

5ddx x

5

5x 6

Make sure the function is written as a power function to use the rule.

The Sum/Difference RuleThe derivative of a sum or a difference of functions is the

sum or difference of the derivatives. In other words, if f and g are both differentiable, then

OR

Objective: Isolate an expressions in order to take the derivative with the Power and Constant Multiple Rules.

ddx f x g x d

dx f x ddx g x

ddx f x g x d

dx f x ddx g x

Example 1If find if

and .

23 h xk x f x ' 5k 5 10, ' 5 3, 5 2,f f h ' 5 16h

Find the derivative of k

2' 3 h xddxk x f x

12' 3 d d

dx dxk x f x h x

12' 3d d

dx dxk x f x h x

12' 3 ' 'k x f x h x

Evaluate the derivative of k at x = 5

' 5k

123 3 16

123 ' 5 ' 5f h

17

Example 2Evaluate:

ddx x

8 12x 5 4x 4 10x 3 6x 5

ddx x

8 ddx 12x 5 d

dx 4x 4 ddx 10x 3 d

dx 6x ddx 5

ddx x

8 12 ddx x

5 4 ddx x

4 10 ddx x

3 6 ddx x d

dx 5

8x 8 1 125x 5 1 44x 4 1 103x 3 1 61x1 1 0

8x 7 60x 4 16x 3 30x 2 6

Sum and Difference Rules

Constant Multiple Rule

Power Rule

Simplify

Example 3Find if . 2 3 4f x x 'f x

12 7d ddx dxx

First rewrite the absolute value function as a piecewise function

Since the one-sided limits are not equal, the derivative does not exist at the vertex

32

32

2 3 4 if

2 3 4 if

x xf x

x x

2 7ddx x

1 12 1 0x

2

Find the Left Hand Derivative

2 7d ddx dxx

12 1d ddx dxx

2 1ddx x

1 12 1 0x

2

Find the Right Hand

Derivative 2 1d ddx dxx

32

32

2 7 if 2 1 if x xx x

32

32

32

2 if ' if

2 if

xf x x

x

Example 4Find the constants a, b, c, and d such that the graph of

contains the point (3,10) and has a horizontal tangent line at (0,1).

2f x ax bx c

What do we know:1. f(x) contains the points (3,10) and (0,1)

2. The derivative of f(x) at x=0 is 0

210 3 3a b c

Use the points to help find a,b,c

21 0 0a b c

210 3 3 1a b 1 c

10 9 3 1a b 9 9 3a b 3 3a b

We need another equation to find a and b

2' 1ddxf x ax bx Find the Derivative

2' 1d d ddx dx dxf x ax bx

2 1' 1d d ddx dx dxf x a x b x

2 1 1 1' 2 1 0f x a x b x

' 2f x ax b We know the derivative of f(x) at x=0

and x=1 is 0

Example 4 (Continued)Find the constants a, b, c, and d such that the graph of

contains the point (3,10) and has a horizontal tangent line at (0,1).

2f x ax bx c

What do we know:1. f(x) contains the points (3,10) and (0,1)

2. The derivative of f(x) at x=0 is 0

0 2 0a b

AND

' 2f x ax b

0 b

1 c3 3a b

Find a

3 3a b Use the Derivative

3 3 0a 1 a

a = 1, b = 0, and c = 1

Derivative of CosineUsing the definition of derivative, differentiate f(x) = cos(x).

cosddx x cos cos

0lim x h x

hh

cos cos sin sin cos

0lim x h x h x

hh

cos 1 cos sin sin

0lim x h x h

hh

cos 1 cos sin sin

0 0lim limx h x h

h hh h

1 cos sin

0 0cos lim sin limh h

h hh hx x

cos 0 sin 1x x

sin x

Find the derivative of

sine if you have time

Derivative of SineUsing the definition of derivative, differentiate f(x) = sin(x).

sinddx x sin sin

0lim x h x

hh

sin cos sin cos sin

0lim x h h x x

hh

sin 1 cos sin cos

0lim x h h x

hh

sin 1 cos sin cos

0 0lim limx h h x

h hh h

1 cos sin

0 0sin lim cos limh h

h hh hx x

sin 0 cos 1x x

cos x

Derivatives of Sine and Cosine

ddx sin x cos x

ddx cos x sin x

Example 1Differentiate the function:

ddx

12 cos x 5 sin x x 7 3

12

ddx cos x 5 d

dx sin x ddx x

7 3

12 sin x 5 cos x 7

3 x7 3 1

sinx2 5 cos x 7

3 x4 3

Sum and Difference Rules

Constant Multiple Rule

Power Rule AND Derivative of Cosine/Sine

Simplify

f x cosx2 5 sin x x 73

ddx

12 cos x d

dx 5 sin x ddx x

7 3

Rewrite the ½ to pull it out easier

Rewrite the radical to use the power rule

Example 2Find the point(s) on the curve where the tangent line is horizontal.

y x 4 8x 2 4

y' ddx x

4 8x 2 4

y' ddx x

4 ddx 8x 2 d

dx 4

y' ddx x

4 8 ddx x

2 ddx 4

y'4x 4 1 82x 2 1 0

Horizontal Lines have a slope of Zero.

First find the derivative.

Find the x values where the derivative (slope) is zero

y'4x 3 16x

0 4x 3 16x

0 4x x 2 4

0 4x x 2 x 2

x 0, 2, 2Find the corresponding y

values

y 0 4 8 0 2 4

y 2 4 8 2 2 4

y 2 4 8 2 2 4

4

12

12

0,4 2, 12 2, 12

Calculus Synonyms

The following expressions are all the same:

• Instantaneous Rate of Change• Slope of a Tangent Line• Derivative

DO NOT CONFUSE AVERAGE RATE OF CHANGE WITH INSTANTANEOUS RATE OF CHANGE.

Position FunctionThe function s that gives the position (relative to the origin)

of an object as a function of time t. Our functions will describe the motion of an object moving in a horizontal or vertical line.

1 2 3 4 5 6

2

-2

-4

s(t)

t Origin

2

-2

-4

TIME:10234567

Description of Movement:

UpwardDownwardNo Movement

DisplacementDisplacement is how far and in what direction something is from

where it started after it has traveled. To calculate it in one dimension, simply subtract the final position from the initial position. In symbols, if s is a position function with respect to time t, the displacement on the time interval [a,b] is:

1 2 3 4 5 6

2

-2

-4

s(t)

t

s s b s a EX: Find the displacement between time 1 and 6.

6 1s s 4 2 6

Average VelocityThe position function s can be used to find average velocity

(speed) between two positions. Average velocity is the displacement divided by the total time. To calculate it between time a and time b:

1 2 3 4 5 6

2

-2

-4

s(t)

t

s a s bst a b

EX: Find the average velocity between time 2 and 5.

5 25 2

s s 4 2

5 2

2

Two Points Needed

It is the average rate of change or slope.

Instantaneous VelocityThe position function s can be used to find instantaneous

velocity (often just referred to as velocity) at a position if it exists. Velocity is the instantaneous rate of change or the derivative of s at time t:

1 2 3 4 5 6

2

-2

-4

s(t)

t

0

lim 's t t s ttt

v t s t

EX: Graph the object’s velocity where it exists.

1 2 3 4 5 6

2

-2

-4

v(t)

t

Slope = 2

Corner Slope = 0

Slope = -4

Slope = -2Slope = 0

Slope = 4

One Point Needed

Position, Velocity, …Position, Velocity, and Acceleration are related in the

following manner:

Position:

Velocity:

( )s t

'( ) ( )s t v t

Units = Measure of length (ft, m, km, etc)

The object is…Moving right/up when v(t) > 0Moving left/down when v(t) < 0Still or changing directions when v(t) = 0

Units = Distance/Time (mph, m/s, ft/hr, etc)Speed = absolute value of v(t)

Example 1The position of a particle moving left and right with respect to an origin is

graphed below. Complete the following:1. Find the average velocity between time 1 and time 4.2. Graph the particle’s velocity where it exists.3. Describe the particles motion.

1 2 3 4 5 6

2

-2

-4

s(t)

t

Example 2 Sketch a graph of the function that describes

the motion of a particle moving up and down with the following characteristics:

The particle’s position is defined on [0,10]

The particle’s velocity is only positive on (4,7)

The average velocity between 0 and 10 is 0.

Example 3The position of a particle is given by the equation

where t is measured in seconds and s in meters.

(a) Find the velocity at time t.

s t t 3 6t 2 9t

s' t v t ddt t

3 6t 2 9t

v t ddt t

3 ddt 6t 2 d

dt 9t

v t ddt t

3 6 ddt t

2 9 ddt t

1

v t 3t 3 1 62t 2 1 9t1 1

v t 3t 2 12t 9

The derivative of the position function is the

velocity function.

Example 3 (continued)The position of a particle is given by the equation

where t is measured in seconds and s in meters.

(b) What is the velocity after 2 seconds?

(c) What is the speed after 2 seconds?

s t t 3 6t 2 9t

v 2

3 2 2 12 2 9

3

v 2

3

m/s

3 m/s

Example 3 (continued)The position of a particle is given by the equation

where t is measured in seconds and s in meters.

(d) When is the particle at rest?

s t t 3 6t 2 9t

0 3t 2 12t 9

0 3 t 2 4t 3

0 3 t 1 t 3

t 1, 3

The particle is at rest when the velocity is 0.

After 1 second and 3 seconds