3.1 – Derivative of a Function
Slope of the Tangent Line
If f is defined on an open interval containing c and
the limit exists, then
y
f (a  h)  f (a)
lim
 lim
m
h  0 x
h0
h
and the line through (c, f (c)) with slope m is the
line tangent to the graph of f at the point
(c, f (c)).
The Slope of the Graph of a Linear Function
 Find the slope of the graph of
at the point (2, 1).

f (x)  2x  3
The Slope of the Graph of a Nonlinear Function
 Find the slope of the graph of
f ( x)  x  1
2
at the point (0, 1) and (-1, 2).
 Find the equation of the tangent line at each point.


Definition
Derivative – The derivative of f at x is given by
f (x  h)  f (x)
f (x)  lim
h0
h
provided the limit exists. For all x for which this limit
exists, f  is a function of x.
Find the Derivative of the Function
f (x)  x  5x  2
2
Alternate Definition
 The derivative of the function f at the point x = a is
the limit
f (x)  f (a)
f (a)  lim
xa
x a
provided the limit exists.

Find the Derivative of the Functions
f (x)  x
Homework
 p.105 ~ 1-9 (O), 13-16, 17, 19
Reflection
 p.105 ~ 1-9 (O), 13-16, 17, 19
Differentiation Rules
3.3.1 (ALSO 3.2)
When Derivatives Do Not Exist
When Derivatives Do Not Exist
When Derivatives Do Not Exist
When Derivatives Do Not Exist
The Constant Rule
 The derivative of a constant function is 0. That is, if c is a
real number, then
.

d
c   0
dx
The Power Rule
 If n is a rational number, then the function f (x) = xn
is differentiable and
d n
x  nx n1
dx
 
For f to be differentiable at x = 0, n must be a
number such that xn–1 is defined on an interval
containing 0.

Find the Derivative of the Function
yx
f (x)  x
g(x)  x
3
1
y 2
x
3
The Constant Multiple Rule
 If f is a differentiable function and c is a real number, then
cf is also differentiable and
d
cf (x)  c.f (x)

dx

Find the Derivative of the Function
4x
g(x) 
5
3x
y
2

2
2
f (x) 
x
The Sum and Difference Rules
 The sum (or difference) of two differentiable functions f
and g is itself differentiable. Moreover, the derivative of
f + g (or f – g) is the sum (or difference) of the derivatives
of f and g.
d
f (x)  g(x)  f (x)  g(x)

dx

Find the Derivative of the Function
f (x)  x  4x  5
3
4
x
3
g(x)    3x  2x
2
The Slope of a Graph
f (x)  x
x  1 , x  0 , and x  1 .
 Find the slope of the graph of



4
when
The Tangent Line
 Find an equation of the tangent line to the graph
of f (x)  x 2 when x  2 .

The Product Rule
 The product of two differentiable functions f and g is itself
differentiable. Moreover, the derivative of fg is the first
function times the derivative of the second, plus the
second function time the derivative of the first.
d
f (x)g(x)  f (x)g(x)  g(x) f (x)

dx
Find the Derivative
h(x)  3x  2x
2
5  4x 
Find the Derivative
yx
35
y x
3

x
x 3

The Quotient Rule
 The quotient of two differentiable functions f and g
is itself differentiable for all values of x for which
g(x) ≠ 0. Moreover, the derivative of f / g is the
denominator times the derivative of the numerator
minus the numerator times the derivative of the
denominator, all divided by the square of the
denominator.
d é f (x) ù g(x) f ¢(x) - f (x)g¢(x)
,
ê
ú=
2
dx ë g(x) û
[ g(x)]
g(x) ¹ 0
Find the Derivative
5x  2
y 2
x 1
Find the Derivative
1
3
x
y
x5
Homework
 p. 114 ~ 1-10
 p. 124 ~ 1-5, 7-29 (O), 30, 32
Higher Order Derivatives
3.3.2
Which Rule Do I Use?
 Find the Derivative
5x
y
8
y
x  3x
y
6
4
2

3 3x  2x
7x
2


9
y 2
5x
Higher-order Derivative Notation
 First Derivative:
 Second Derivative:
 Third Derivative:
 Fourth Derivative:
 nth Derivative:
Find the Second Derivative
32
f (x)  x  2
x
g(x)  4x
3/2
Instantaneous Rate of Change
 A population of 500 bacteria is introduced into a
culture and grows in number according to the
equation
4t 

P(t)  500  1
2
 50  t 
where t is measured in hours. Find the rate at which
the population is growing when t = 2
Velocity and Other Rates of
Change
3.4
Position and Velocity
 If a billiard ball is dropped from a height of 100
feet, its height s at time t is given by the position
function
2
s  16t 100
where s is measured in feet and t is measured in
seconds. Find the average velocity over each of
the following time intervals.

[1, 2]
[1, 1.5]
[1, 1.1]
Position Function
1 2
s(t)  gt  v0 t  s0
2

Instantaneous Velocity
 At time t = 0, a diver jumps from a platform diving
board that is 32 feet above the water. the position of the
diver is given by
2
s(t)  16t 16t  32
where s is measured in feet and t is measured in
seconds.
 When does
 the diver hit the water?
 What is the diver’s velocity at impact?
Higher-order Derivatives
Because the moon has no atmosphere, a falling object on the
moon encounters no air resistance. In 1971, astronaut David
Scott demonstrated that a feather and a hammer fall at the
same rate on the moon. The position function for each of
these falling objects is given by
s(t)  0.81t  2
2
where s(t) is the height in meters and t is the time in seconds.
What is the ratio of Earth’s gravitational force to the moon’s?
Free Fall
A silver dollar is dropped from the top of a
building that is 1362 feet tall.
 Determine the position, velocity, and
acceleration functions for the coin.
 Determine the average velocity on the interval
[1, 2].
 Find the instantaneous velocities when t = 1
and t = 2.
 Find the time required for the coin to reach
ground level.
 Find the velocity of the coin at impact.
Free-Fall
 A projectile is shot upward from the surface of Earth
with an initial velocity of 120 meters per second.
What is its velocity after 5 seconds?
 What is the maximum height of the projectile?
Homework
 p.135/1, 3, 7, 9-15 (O), 19, 21
Derivatives of Trigonometric
Functions
3.5
The Derivative of Sine
Derivatives of Sine and Cosine Functions
d
sin x  cos x
dx
d
cos x  sin x
dx
Find the Derivative of the Function
y  sin x
y  x  cosx
sin x
y
2

Find the Derivative
y  3x sin x
2
y  2x cos x  2sin x
y  x sin x cos x
2
Simple Harmonic Motion
 A weight hanging from a spring is stretched 5 units
beyond its rest position (x = 0) and released at time
t = 0 to bob up and down. Its position at any later
time t is
What are its velocity and acceleration at time t?
Describe its motion. s = 5cost.
Derivative of Tangent
Derivatives of Trigonometric Functions
d
2
tan x  sec x
dx
d
2
cot x  csc x
dx
d
secx  secx tan x
dx
d
cscx  cscx cot x
dx
Find the Derivative
y  x  tan x
y  cscx  cot x
y  xsecx
Find the Second Derivative
y = x + sin x
2
f (x)  secx
Homework
 p. 146/ 1-9 (O), 11-23 (O), 27-39 (O), 43
The Chain Rule
4.1
The Chain Rule
If y = f (u) is a differentiable function of u and u =
g(x) is a differentiable function of x, then y = f (g(x))
is a differentiable function of x and
dy dy du


dx du dx
or, equivalently,

d
 f (g(x))  f (g(x))g(x).
dx
Identify the inner and outer functions
Composite y = f (g(x)) Inner u = g(x) Outer y = f (u)
1
1. y 
x 1
2. y  sin 2x
3. y 
3x 2  x 1
4. y  tan x
2
The General Power Rule
 If
y = [ u(x)] , where u is a differentiable function of
n
x and n is a rational number, then
dy
n-1 du
= n [ u(x)]
dx
dx
or, equivalently,
d n
n1
u  nu u .

dx
Find the Derivative

y  x 1
2

3
Find the Derivative
f (x)  3x  2x
f (x) 
f (x) 
3
x
2

2 3
1
2
7
2t  3
2
Homework
 Chain Rule Worksheet
Factoring Out the Least Powers
 Find the Derivative
f (x)  x 2 1  x 2
Factoring Out the Least Powers
 Find the Derivative
f (x) 
x
3
x 4
2
Factoring Out the Least Powers
 Find the Derivative
2
3x 1 
y   2

x  3
Find the Derivative
y  sin 2x
y  cos(x 1)
y  tan 3x
Find the Derivative
y  cos(3x )
2
y  (cos 3)x
2
y  cosx
f (t)  sin 4t
3
Trig Tangent Line
 Find an equation of the tangent line to the graph of
f (x)  2sin x  cos 2x
at the point (π, 1). Then determine all values of x in
the interval (0, 2π) at which the graph of f has a
horizontal tangent.

Homework
 p.153/ 1-11odd, 21-39odd, 59
Implicit Differentiation
4.2
Find dy/dx
d
 x 3  
dx
d
 y 3  
dx
d
 x  3y 
dx
d
 xy 2  
dx
Guidelines for Implicit Differentiation
1.
2.
3.
4.
Differentiate both sides of the equation with respect to
x.
Collect all terms involving dy / dx on the left side of the
equation and move all other terms to the right side of
the equation.
Factor dy / dx out of the left side of the equation.
Solve for dy / dx.
Find the derivative
y  y  5y  x  4
3
2
2
Homework
 p.162/ 1-19odd, 49, 51
Example
 Determine the slope of the tangent line to the graph of
x  4y  4
2
 at the point


2
.
 2,1/ 2
Example
 Determine the slope of the tangent line to the graph of
3x  y
2
 at the point


3,1.
  100xy
2 2
Finding the Second Derivative Implicitly
x  y  25
2
2
Finding the Second Derivative Implicitly
y  4x
2
x 2  y 2  36
1 xy  x  y
Example
 Find the tangent and normal line to the graph given by
x x  y   y
2
 at the point


 2 / 2,
2
2

2 / .2
2
Homework
 p.162/ 21-25odd, 27-30, 31-43odd
Inverse Functions
4.3
Definition of Inverse Function
 A function g is the inverse function of the function f if
for each x in the domain of g.
f (g(x))  x
and
for each x in the domain of f.
g ( f ( x))  x
Verifying Inverse Functions
 Show that the functions are inverse functions of
each other.
f (x)  2x 1
3

and
g(x)  3
x 1
2
The Existence of an Inverse Function
1.
2.
A function has an inverse function if and only if it is
one-to-one.
If f is strictly monotonic on its entire domain, then it is
one-to-one and therefore has an inverse function.
Existence of an Inverse Function
 Which of the functions has an inverse function?
f (x)  x  x  1
3
f (x) x  x 1
3
Finding an Inverse
 Find the inverse function of
f (x)  2x  3
.
The Derivative of an Inverse Function
 Let f be a function that is differentiable on an interval I.
If f has an inverse function g, then g is differentiable at
any x for which f (g(x))  0 . Moreover,
1
g(x) 
,
f (g(x))

f (g(x))  0.

Example
1 3
f (x)  x  x .1
4

Let
a)
What is the value of
f (x) when x = 3?
b)
What is the value of
f  (x)


1
1
when x = 3?
Homework
 p. 44/ 1-6, 7-23odd, 43
 p. 170/ 28, 29bc
Inverse Trigonometric
Functions
3.8
The Inverse Trigonometric Functions
Function
y  arcsin x iff sin y  x
y  arccos x iff cos y  x
y  arctan x iff tan y  x
y  arccot x iff cot y  x
y  arcsecx iff secy  x
y  arccscx iff cscy  x
Evaluating Inverse Trigonometric Functions

a)
b)
c)
d)
Evaluate each function.
 1
arcsin   
 2
arccos0
arctan 3
arcsin 0.3
Solving an Equation

arctan 2x  3 
4
Using Right Triangles
a) Given y = arcsin x, where 0  y 



2

b) Given y  arcsec 5 / 2 , find tan y.
, find cos y.
Homework
3.8 Inverse Trig Review worksheet
Derivatives of Inverse Trigonometric Functions
 Find
d
arcsin x
dx
Derivatives of Inverse Trigonometric Functions
d
1
arcsin x 
dx
1  x2
d
1
arccos x   
dx
1  x2
d
1

arctan x 
2
dx
1 x
d
1
arccot x  
dx
1  x2
d
1 
arcsecx  
dx
x x 2 1
d
1
arccscx   
dx
x x 2 1
Differentiating Inverse Trigonometric Functions
a)
d
arcsin  2 x  
dx
b)
d
arctan  3 x  
dx
c)
d 

arcsin
x

dx 
d)
d
arc sec  x 2  

dx 
Differentiate and Simplify
y  arcsin x  x 1  x
2
Homework
 p. 170/ 1-27odd, 31ab
Derivatives of Exponential
and Logarithmic Functions
4.4
Properties of Logarithms
If a and b are positive numbers and n is rational, then
1.
ln 1  0
2.
ln ab   ln a  ln b
3.
4.
 
ln a n  n ln a
a 
ln    ln a  ln b
b 
Expand the following logarithms
10
ln  
 9 
6x 
ln  

 5 
ln 3x  2
ln
x
x
2
3

3
2
x 1
2
Solving Equations
 Solve 7 = ex + 1
 Solve ln(2x  3) = 5
Derivative of the Natural Exponential Function
 
d x
e
dx
 
d x
e  ex
dx
Examples
d 2x 1
e

dx
 
d 3 / x
e

dx


Derivatives for Bases Other than e
 
d x
a
dx
 
d x
a  ln a a x
dx
Find the derivative of each function.
f ( x)  2
y2
x
3x
2t
3
g (t ) 
t
Homework
p.44/ 33-38
p.178/ 1-13odd, 29, 30
Derivative of the Natural Logarithmic Function
d
ln x 
dx
d
1
ln x  ,
dx
x
x0
Example
 Find the derivative of ln (2x).
Find the Derivative
1. ln x 2  1
2. xln x



3. ln x 
3
Derivatives for Bases Other than e
d
log a x 

dx
d
1
log a x 

dx
ln ax
Example
 Differentiate

y  log 10 cosx
.
Example
 Differentiate

f (x)  ln x .1
Example
x ( x +1)
2
 Differentiate
f (x) = ln
.
2
2x -1
3
Find an equation of the tangent line to the graph at the
given point.
x  y  1  ln  x  y  ,
2
2
1, 0 
Homework
 p. 178/15-27odd, 37-41odd
Comparing Variable and Constants
 
d e
e
dx

d 3 x 2  log4 7
e
dx
 
d e
x
dx

Derivative Involving Absolute Value
d
u
ln u  
dx
u
Find the derivative
f (x)  ln cos x
Use implicit differentiation to find dy/dx.
 ln xy + 5x = 30
Logarithmic Differentiation
 Differentiate
x  2

f ( x) 
2
x2  1
, x  2.
Logarithmic Differentiation
 
d x
x
dx
More Logarithmic Differentiation
 
d x 1
x
dx

d
1/ x
1 x 
dx

Homework
 p. 179/ 31, 33-36, 43-55odd