INTRODUCTION TO

DIFFERENTIATION

DEFINITION OF A DERIVATIVE The derivative of a function f(x),

denoted f’(x) is the slope of a tangent line to a curve at any given point.Or the slope of a curve at any given pointOr the instantaneous rate of change of y

with respect to x.

What is a tangent line?

ESTIMATING DERIVATIVES Find the slope of the tangent line to the

graph of f at the given x value

1. f(x) = x2 at x = 2

2. f(x) = x2 + 2x at x = -2

Answer to 1) f’(2) = 4 Answer to 2) f’(-2) = -2

WHAT IS A DERIVATIVE FUNCTION? The derivative function for example 1

is…

f’(x) = 2x f’(2) = 4

The derivative function for example 2 is…

f’(x) = 2x + 2 f’(-2) = -2

BASIC RULES OF DIFFERENTIATION

Rules to find functions that will give you the slope of the tangent lines to a curve at any

value of x

CONSTANT RULE The derivative of any constant is 0.

That is….

If ( ) then '( ) 0f x k f x

or 0dk

dx

POWER RULE For any real number ‘n’

1If ( ) then '( )n nf x k x f x kn x

EXAMPLES USING POWER RULEFind the derivative of each function below.

51. ( ) 3f x x

5 22. ( ) 3 4 5f x x x

53. ( ) 3 2f x x

EQUATIONS OF TANGENT LINES Find the equation of the tangent line to

the graph of f at the specified value of x.

f(x) = 3x4 – 2x + 1 at x = 0

Find the value(s) of “x” where the function has a horizontal tangent line

EXAMPLES USING POWER RULE(WITH SIMPLIFICATION)

Find the derivative of each function below.

5/ 2 44. ( ) 3 2f x x x

3 25. ( ) 3 4 5f x x x

4

3 26. ( )f x

x x

EXAMPLES USING POWER RULE(WITH SIMPLIFICATION)

Find the derivative of each function below.

27. ( ) 2f x x

23 4 68. ( )

x xf x

x

5

4

3 2 49. ( )

2

xf x

x x

SPECIAL RULES Derivatives of sine and cosine

If ( ) sin then '( ) cosf x k x f x k x

If ( ) cos then '( ) sinf x k x f x k x

SPECIAL RULES Derivatives of natural logarithm and ex

If ( ) then '( )x xf x k e f x k e

1If ( ) ln then '( )f x k x f x k

x

EXAMPLES OF SPECIAL RULES

1. ( ) 3sin 2cosf x x x

2. ( ) sin 2 xf x x e

23. ( ) 3 4cosf x x x

EXAMPLES OF SPECIAL RULES

4. ( ) 3ln 2f x x

15. ( ) cos sin ln

2f x x x x

26. ( ) 3 xf x x e

HIGHER ORDER DERIVATIVES DERIVATIVES OF DERIVATIVES THE RATE OF CHANGE OF A RATE OF

CHANGE

NOTATION: If y or f(x)1st derivative y’ or f’(x)2nd derivative y’’ or f”(x)3rd derivative y’’’ or f’’’(x)4th or more y(4) or f(4)(x)

EXAMPLES OF HIGHER: Find the 1st and 2nd derivatives of each.

1. ( ) sin 2 xf x x e

5 22. ( ) 3 4 5f x x x

3. ( ) 3ln 2f x x

DERIVATIVES AS A RATE OF CHANGE

EXAMPLE #1 The position of a car is given by the values in the following table.

t(sec) 0 1 2 3 4 5

s(feet) 0 10 32 70 119 178

a) Graph the position function on graph paper

b) Use your graph to find the average velocity for the time periods

below i) 2 to 5 seconds ii) 2 to 3 seconds

c) Estimate the instantaneous velocity when t = 2

ANSWER178 32

5 2avgv

70 32

3 2avgv

OBSERVATIONS:

An average rate of change of a function is ALWAYS found by finding the slope of a secant line from time “a” to time “b”.

An Instantaneous rate of change of a function is found by finding the slope of the tangent line at time “a”. (the derivative!)Sometimes an instantaneous rate can be

estimated with an average rate if an equation is not known!

EXAMPLE #2A ball is dropped from the top of a tower 450 meters above the ground. Its distance

after ‘t’ seconds is given by the equation

Find:a) The average velocity from t = 1 to t =

3 seconds

b) The instantaneous velocity at t = 5 seconds.

2( ) 4.9 450s t t

ANSWER #2a)

b)

1 2

1 2avg

s t s tv

t t

405.9 445.1

3 1avgv

39.219.6 m/s

2avgv

'(5) 49 m/ss

RATE OF CHANGE AND HIGHER DERIVATIVES: A particle moves along a line such

that after ‘t’ seconds its position is given by the equation

a) Find the velocity of the particle at exactly 2 seconds.

b) Find the acceleration at 1 second.

3 2 s( ) 16 100 200 ftt t t

PRODUCT RULE Suppose that f and g are differentiable

functions. Then….

If ( ) ( ) ( )

then '( ) '( ) ( ) ( ) '( )

h x f x g x

h x f x g x f x g x

If ( ) then '( ) ' 'h x f s h x f s f s

EXAMPLES OF THE PRODUCT RULE

1. ( ) 2sin xf x x e

22. ( ) 3ln 2f x x x

23. ( ) 3 4 xf x x e

QUOTIENT RULE Suppose that f and g are differentiable

functions and g(x) 0. Then….

2

( )If ( )

( )

( ) '( ) ( ) '( )then '( )

( )

f xh x

g x

g x f x f x g xh x

g x

2

If ( )

' 'then '( )

nh x

dd n n d

h xd

EXAMPLES OF THE QUOTIENT RULE

2sin1. ( )

x

xf x

e

2

3ln2. ( )

2

xf x

x

123. ( ) 3 4 2f x x x

TRIGONOMETRIC FUNCTIONS If y = sin x y’ = cos x

If y = cos x y’ = -sin x

If y = tan x y’ = sec2 x

If y = cot x y’ = -csc2 x

If y = sec x y’ = sec x tan x

If y = csc x y’ = -csc x cot x

CHAIN RULE Rule for taking the derivative of

functions that are a composition of two or more functions.y= f(g(x))

( ) '( ) '( )( ) ( )g x gdf f g

xx x

d

USING THE CHAIN RULE WITH POWER FUNCTIONS Find the derivative of each function.

531. ( ) 4 7f x x

22. ( ) 2 5f x x x

83 43. ( ) 2 7 4 8k x x x x

USING THE CHAIN RULE WITH PRODUCTS AND QUOTIENTS

Find the derivative of each function.

331. ( ) 4 7 3 7f x x x

3

2

72. g( )

4 2

xx

x

USING THE CHAIN RULE WITH THE NATURAL LOG AND EXPONENTIAL

FUNCTION

Find the derivative of each function.

51. ( ) lnf x x 22. ( ) ln 4 8f x x

sin3. ( ) xf x e 324. ( ) xf x x e

USING THE CHAIN RULE WITH TRIGONOMETRIC FUNCTIONS Find the derivative of each function.

31. ( ) sinf x x 32. g( ) tanx x

4 23. ( ) cos 3 1k x x

WHAT IS THE DIFFERENCE BETWEEN THE FOLLOWING TWO EXPRESSIONS?

1. f ( ) tan sinx x

2. g( ) tan sinx x x