Derivatives
Unit 1
This Unit covers derivatives from the definition of derivative through the power rule, the product rule and
the quotient rule. It also covers tangent lines and rates of change.
The videos we watch in class can be found at: http://www.calculus-help.com/tutorials
Some of the inclass examples can be found at: http://archives.math.utk.edu/visual.calculus/
All of the inclass powerpoints can be found at: http://www.hatboro-horsham.org/evans
1
Derivatives
Evaluate each limit.
1.
2.
3.
3
3

lim x  x x
x 0
x
lim
x 0
x  x  5  x  5
x
( x  x )3  x 3
lim
x 0
x
 x  x   3  x  x    x3  3x 
3
lim
4.
x 0
x
2
Derivative
Derivative
We wished to find the slope of the tangent line to the
graph of a function at a point.
The value of h gets
closer to zero.
Because we only knew one point on the tangent line, we
could not use the slope formula from Geometry.
But we could use the slope formula with a secant line that
passed through the tangent point and another point on
the curve near the tangent point.
We now define the
slope of the tangent
line at point P (x, f(x))
as follows:
As we moved this second point closer to the tangent point,
the position of the secant line became a better
approximation of the tangent line and thus, the slope of
the secant line became a better approximation of the slope
of the tangent line.
mtan  lim
h 0
 msec 

lim
h 0
f ( x  h)  f ( x)
h
Derivative
Derivative
The instantaneous rate of change of a function (the
slope of the tangent line) also has a very special name
in Calculus.
The difference quotient measures the average rate of
change between two points. It is the slope of a secant
line to a graph.
It is called the derivative of the function at x.
The limit of the difference quotient as h approaches
zero measures the instantaneous rate of change at the
point of tangency. It is the slope of the tangent line to
a graph at the point of tangency.
derivative of f(x)= lim
h 0
f ( x  h)  f ( x )
h
There are several ways to represent the derivative of
function f(x). Two of the most common are:
It is the slope of the tangent line to a graph at the
point of tangency.
df
dx
f ( x) and
Derivative
Derivative
We can do this in four steps:
Let’s calculate the slope of the tangent line to the
function that we started with earlier in the exploration:
f ( x )  x3  6 x  2
1. Compute f(x + h).
2. Form the difference f(x + h) – f(x).
We need to find the instantaneous rate of change
of f(x). We need to find the derivative of f(x).
We need to calculate
lim
h 0
3. Form the quotient
f ( x  h)  f ( x )
h
4. Compute f’(x) =
f ( x  h)  f ( x )
h
lim
h 0
.
f ( x  h)  f ( x )
h
.
When you get comfortable with the process and the Algebra required in it, you can
combine the first three steps into one step. For now, we’ll take small steps.
3
Examples:
4
Use the same procedure as the previous exercises. This is merely a different notation for the same
process.
lim
Find the x  0
f ( x  x)  f ( x)
x
of the following functions.
1.
f ( x)  4 x  2
2.
f ( x)  x 2  3x
3.
f ( x)  3x 2  5 x  2
4.
f ( x)  x  2
5.
2
f ( x) 
x 1
5
Discovering the Power Rule
1.
Enter the following in your graphing calculator
y1  x 2
y2
2.
x  h


2
 x2
h
Store the value of h as a number approaching 0, such as “.000000001”
3. Graph
y2 , look at the table of values.
4. With your group, come up with a rule for
y2
5. Now do the same thing and try to predict the following derivatives.
a.
f ( x)  x 3
b.
f ( x)  x 4
c.
f ( x)  x 5
_________________________
_________________________
_________________________
3
2
d.
f ( x)  x
e.
f ( x)  x n
_________________________
_________________________
What is your conjecture for the Power Rule? _________________________________________
6
Practice Set #1
1.
y  3cx 4  4c 2 x  8c
Let
a. Find
dy
[assume c is a constant]
dx
b. Find
dy
dx x 2
c. Find
dy
[assume x is a constant]
dc
2. Differentiate the following.
Function
Rewrite/Simplify
y

Differentiate

x  4x  x  x2 
x2  2x
y
x
y
7
3x3
y
x x 4
2 x
7
Simplify
3.
Differentiate the following and simplify your final answers.
1
2 x3  4 

3
a.
y
b.
g ( x) 
3x  1
3x
f ( x)   x  4 
5
c.
2
4. Find the first and second derivatives of
5. Given f ( x) 
y  3 x2
1 3
x  x 2  x  1 Find the points of the graph where the slope is:
3
a. 0
b. 1
c. 4
8
Practice Set #2:
1.
Find the equation of the line tangent to the curve
2. Find the equation of the line tangent to the curve
y  3x 2  7 x  3 at the point (3,11).
y  2 x3  5 x 2  4 x  1 at the point on the
curve with an abscissa (x value) of 2.
3. If
f ( x)  5x3  4 x 2  7 x  2 , find f '(2)
4. Find the equation of the line that has a slope of 7 and is tangent to the curve,
5. Find the equation of the line tangent to the curve
y  x2  5x  3
y  x3  1 at the point on the curve located 7
units below the x-axis
6. Find the coordinates of the only second quadrant point on the curve
y  x3  7 x 2  5 x  1
where the line tangent to the curve will be parallel to the x-axis.
7. If
y  x3  4 x 2  9 x  3 , find the equation of the line tangent at the point where the curve
crosses the y-axis.
8. If
1
y  x3  2 x 2  3x  7 , find all values of x for which the slope of the curve is positive.
3
9. Line AB is given by
y  9 x  12 . Find the equation of the line tangent to the curve
y  x 2  x  4 and is also parallel to the line AB.
10. Find the equation of the line that has an angle of inclination of
y  9  7 x  x2
9
45o and is tangent to
Conclusions/Applications
Two kinds of rate of change
Q: A car travels 110 miles in 2 hours. What’s its average
rate of change (speed)?
A: 110/2 = 55 mi/hr. That is, if we drive 55 miles in an hour,
then in 2 hours, we will have driven 110 miles.
Q: If you are driving and suddenly look at your odometer,
which says 60 mi/hr, what kind of rate of change is that?
A: Instantaneous R.O.C. That is, the rate at that particular
time instance.
Average R.O.C. is over a period of time
Instantaneous R.O.C. is at a given point of time.
• Yes! That’s right, it’s 0! Namely, the slope
of the tangent line is 0! But a line with 0
slope is a horizontal line.
• Thus, a maximum/minimum point has
horizontal tangent line.
• To find the maximum/minimum point of a
function f(x), we look for the point x where
its tangent line has slope 0.
c
Important Idea
The average rate of change
(speed) over a time period is
the slope of the secant line
connecting the beginning
and end of the time period.
Average y
y y
 2 1
Rate of 
t
t2  t1
Change
Important Idea
The instantaneous
velocity at a point, or any
other rate of change, is
the slope of the tangent
line at the point
Rate of Change (Notes)
The derivative can be used to determine the rate of change of one variable with respect to another.
Ex: Population growth, production rates, rate of water flow, velocity and acceleration.
Ex: Free fall Position function. A function, s, that gives position (relative to the origin) of an
object as a function of time.
s
1 2
gt  vot  so
2
Consider: A ball dropped from a 160 foot building.
Note:
Rate 
Dis tan ce
time
10
Therefore, the average velocity is
change in dis tan ce y

change in time
x
Find the average velocity over the given time intervals:
a. [1,2]
b. [1,1.5]
c. [1,1.1]
Note: Negative velocity indicates _________________
Generally if, s = s(t) is the position for an object moving in a straight line, then the velocity of the
object at time t is:
s (t  t )  s (t )
 s '(t )
t 0
t
v(t )  lim
Find the instantaneous velocity when t = 1.1 sec.
Position Function:
Velocity Function:
Acceleration Function:
11
Example Set:
1.
A stone is thrown vertically upward from the ground with an initial velocity of 32 ft/sec, and
an equation of motion
s  16t 2  32t
. Find
a. The average velocity of the stone during the time interval
b. The instantaneous velocity of the stone at
3
5
t 
4
4
3
5
sec and at sec
4
4
c. How many seconds will it take for the stone to reach its highest point?
d. How high will the stone go?
e. How many seconds will it take the stone to reach the ground?
f.
The instantaneous velocity of the stone when it reaches the ground.
2. At t = 0, a diver jumps from a diving board that is 32 ft above the water. The position of the
diver is given by the equation
s(t )  16t 2  16t  32
sec.
a. When does the diver hit the water?
b. What is the diver’s velocity at impact?
12
where s is measured in ft and t is
Practice Set #3 - Rate of Change
1.
A company finds that charging q dollars per unit produces a monthly revenue, R,
R  12000q  1000q 2 , 0  q  12
2.
Find the rate of change of R with respect to q when q = 5.
If the effectiveness, E, of a painkilling drug t hours after entering the blood stream is given by
E
1
t 3  3t 2  9t  , 0  t  4 Find the rate of change of E with respect to t when t = 1 ; t = 3 ; t = 4

27
3.
A diver dives from a 20-foot platform. Her initial velocity is 4 feet per second. What is her velocity when
she hits the water?
4.
An astronaut standing on the moon throws a rock into the air. The height of the rock is given by
s
27 2
t  27t  6
10
where s is measured in feet and t is measured in seconds. Find the acceleration
of the rock and compare it with the acceleration due to gravity on earth.
5.
A ball is thrown upward from ground level, and its height is given by
s(t )  16t 2  48t
where s is measured in feet and t is measured in seconds.
a. Write an expression for the velocity and acceleration of the ball.
b.
After how many seconds will the ball reach its maximum height and how high will it be at that
time?
c.
6.
What is the velocity of the ball as it hits the ground?
A balloonist drops a sandbag from a balloon 160 feet above the ground.
a.
Find the velocity of the sandbag after 1 second.
b.
With what velocity does the sandbag hit the ground?
13
7.
A projectile is fired directly upward from the ground with an initial velocity of 112 ft/sec.
a. What is the velocity at 3 seconds?
b.
What is the maximum height that the projectile will reach?
c.
What is the velocity at the instant that the projectile strikes the ground?
8.
A pebble is dropped from a height of 5184 feet. Find the pebble’s velocity when it hits the ground.
9.
A ball is thrown straight down from the top of a 220-foot building with an initial velocity of -22 feet per
second.
a. What is it velocity after 3 seconds?
b.
What is its velocity after falling 121 feet?
10. To estimate the height of a building, a stone is dropped from the top of the building. How high is the
building if it strikes the ground 6.8 seconds after it is dropped?
11. A ball is dropped from a height of 100 feet. One second later another ball is dropped from a height of 75
feet.
a. Which ball hits the ground first?
b.
How fast is it going when it hits the ground?
12. A man standing on top of 256 foot building throws a ball straight up in the air at a rate of 96 feet per
second.
a. What is its average velocity for the first two seconds?
b.
How fast is it going after 2.5 seconds?
c.
What is the highest point the ball will reach?
d.
How fast is the ball traveling when it hits the ground?
14
Practice Set #4 - Rate of Change
1. A ball is thrown upward from ground level, and its height is given by s(t )  16t  64t
2
where s is measured in feet and t is measured in seconds.
a. Write an expression for the velocity and acceleration of the ball.
b. After how many seconds will the ball reach its maximum height and how high will it be
at that time?
c. What is the average rate of change in the time interval: [1, 1.4]
d. What is the velocity when t = 1 second?
e. What is the velocity the instant that the ball hits the ground?
2. If the effectiveness, E, of a painkilling drug t hours after entering the blood stream is given by
E
1
t 3  5t 2  15t  , 0  t  4 Find the rate of change of E with respect to t when

25
a.
b.
c.
t=1
t=2
t=4
3. A balloonist drops a sandbag from a balloon 240 feet above the ground.
a. Find the velocity of the sandbag after 2 seconds.
b. With what velocity does the sandbag hit the ground?
c. What is the average rate of change in the time interval: [2, 3.5]?
15
4. To estimate the height of a building, a stone is dropped from the top of the building. How high is
the building if it strikes the ground 7.4 seconds after it is dropped?
5. A man standing on top of 128 foot building throws a ball straight up in the air at a rate of 96 feet
per second.
a. What is its average velocity for the first two seconds?
b. How fast is it going after 2.4 seconds?
c. What is the highest point the ball will reach?
d. How fast is the ball traveling when it hits the ground?
6. A ball is thrown straight down from the top of a 144 foot tall building at a speed of 48 feet per
second.
a. What is the velocity of the ball at 1.5 seconds?
b. How fast is the ball going when it hits the ground?
16
Practice Set #5 - Rate of Change
1.
A toy rocket is ejected from the top of a building 192 feet above the ground with an initial
velocity of 64 feet per second.
a. What is the highest level the rocket will reach?
b. After how many seconds will it reach the ground?
c. What is the velocity of the rocket at the instant it touches the ground?
2. A ball is thrown upward with an initial velocity of 32 feet per second.
a. At what instant will the ball be at its highest point and how high will it rise?
b. What is the velocity of the ball at 2 seconds?
3.
A water filled balloon is dropped from a height of 224 feet.
a. When will the balloon hit the ground and what will the velocity of the balloon be the
instant it hits the ground?
b. What is the average velocity of the balloon after 2 seconds?
c. What is the instantaneous velocity by at 1.5 seconds?
4.
To estimate the depth of a well a stone is dropped into the well. The stone hits the bottom of the
well 5 seconds after it is dropped. How deep is the well?
17
5. A projectile is fired directly upward from the top of a building 160 feet high with an initial
velocity of 112 feet per second.
a.
What is the velocity at 2 seconds?
b. What is the maximum height that the projectile will reach?
c. If the projectile falls back onto the roof, what will be the velocity when it hits the roof?
d. If the projectile falls to the ground instead, what will the velocity be when it hits the
ground?
e. How long will it take to hit the ground?
6. A spherical balloon is being blown up. The volume of a sphere is given by the formula
4
V   r 3 Find the rate of change of the volume with respect to the radius and find the rate of
3
change of the volume of the balloon when the radius is at 3 inches.
18
Example Set - Higher Order Derivative Problems.
1.
f ( x)  ax 2  bx  c, f (1)  5 ; f '(1)  3 ; f ''(1)  4 ; find f (2)
2.
f ( x)  x3  ax 2  bx  c, f (1)  9 ; f '(2)  22 ; f ''(3)  24 ; find f (2)
3.
f ( x)  x3  x 2  5x  1, find f '( x) when f ''( x)  0
f ( x)  x3  3x 2  5x  7, find f '( x) when f ''( x)  0
19
product rule:
d
dv
du
 uv   u  v
dx
dx
dx
Notice that this is not just the
product of two derivatives.
This is sometimes memorized as:


 
d  uv   u dv  v du
d  2
3
x

3
2
x
 5 x   x 2  3  6 x 2  5    2 x3  5 x   2x 

dx

d
2 x5  5 x3  6 x3  15 x
dx

d
2 x 5  11x 3  15 x
dx
10 x 4  33 x 2  15



6 x 4  5 x 2  18 x 2  15  4 x 4  10 x 2
10 x 4  33 x 2  15

20
Examples
21
quotient rule:
du
dv
u
d u
dx
dx
 
2
dx  v 
v
 u  v du  u dv
d 
v2
v
v
or
x 2  3 6 x 2  5    2 x3  5 x   2 x 
d 2 x3  5 x


2
dx x 2  3
x2  3



Examples
22
Practice Set #6.
1. Let
f ( x)  3x 2  14 x   x8 . Find f '( x ).
2. Let y  9 x 
3. Let f (t ) 
4. Find
dy
1

7

x
.
Find
dx
x3
2
 7t 3  5t . Find f '(t ).
3
16 t
d  
4 
2
x

.
dx 
x
2
3
5. Let
y  6 x  x 2  e . Find
6. Find
d  x3  4 x 4  6 

.
dx 
x2

7. Differentiate the function
8. Let f ( x) 
dy
dx
1

f ( x)   3 x8  1  4 x 2   .
x

3x  1
. Find f '( x ).
x  x4
2


9. Find the derivative of the function g (t )   2t 
23
1 3 2
 t .
t2 
10. Let y 
11. Find
dy
2 x  5x2
. Find
dx
4x 1
d  2 x  7 x 

.
dx 
4

12. Write an equation for the line tangent to
13. Find any points at which the curve y 
14. At what value(s) of x does the curve
y  x3  x
at the point (1,2)
1 3
x  4 x 2  12 x has a horizontal tangent line.
3
y  x2  8x
4x  y  7 ?
24
have at tangent line parallel to the line
Summary of trig derivatives
d
sin x  cos x
dx
d
cot x   csc 2 x
dx
d
cos x   sin x
dx
d
sec x  sec x  tan x
dx
d
tan x  sec 2 x
dx
d
csc x   csc x  cot x
dx

Examples:
25
Practice Set #7 - Trig Function Derivatives
Find
f '( x)
if
f ( x)  7 x cot x
2. Find
f '( x)
if
f ( x)  5 tan x sec x
3. Find
f '( x)
if
f ( x) 
4. Find
 
f ''  
6
5. Find
 3 
f ' 
 4 
6. Find
 
f ' 
4
1.
if
if
if
4 cot x
5x
f ( x)  4sin x
f ( x)  2 cot x
f ( x)  3sin x tan x
7. Find the equation of the line tangent to the curve f(x) when
x

3
if
f ( x)  2 csc x
26
Practice Set #8 - Trig Function Derivatives
Find
f '( x)
if
f ( x)  6 x sec x
2. Find
f '( x)
if
f ( x)  cos x sec x
3. Find
f '( x)
if
f ( x) 
4. Find
f ''  x 
5. Find
 
f ' 
4
if
f ( x)  2 csc x
6. Find
 
f ' 
3
if
f ( x)  3sin x tan x
1.
if
2 tan x
3x
f ( x)  cot x
7. Find the equation of the line tangent to the curve f(x) when
27
x

6
if
f ( x)  sec x
Practice Set #9 - Higher Order Derivative Problems.
1.
If f ( x)  4 x3  5x 2  2 x  6, find f ''(3)
2.
 
If f ( x)  2sin x, find f ''  
4
3.
 
If f ''( x)  2 x sin x, find f '''  
6
4.
f ( x)  x3  3x 2  5x  7,
Find the equation of the line tangent to the curve
when f’’(x) = 6
28
Review
1.
Find the equation of the line tangent to the curve
f ( x)  x3  3x 2  x,
at the point
(3,3)
2. Find the equation of the line that has a slope of 5 and is tangent to the curve,
f ( x)  8  13x  3x 2
3.
Find the equation of the line tangent to the curve
3x  2
x 1
4. If
f ( x) 
5. If
f ( x)  x 2  6 x  5,
y x
at the point where x = 9
find the value of f’(6)
find the only point on the curve where the tangent line will be
parallel to the x-axis.
6. Find the equation of the line tangent to the curve
y
6
x
at the point on the curve
located 3 units above the x-axis.
7. Find the equation of the line tangent to
the curve crosses the y-axis.
29
y  3x3  6 x 2  2 x  4,
at the point where
8. Find the derivative of
3x 2  5
y
4x 1
9. Find the equation of the line tangent to the curve
y
1
x
y  x3  3x 2  24 x
10. For what values of x will the curve
11. Find the equation of the line with an angle of inclination of
curve
y
14. If
have a negative slope?
45o
that is tangent to the
4
x2
12. Find the point on the curve
13. The line
at the point where x = 4.
y  8 x  11
y  12 x
is tangent to
where the slope of the curve is 2.
y  3x 2  4 x  1,
Find the point of tangency.
f ( x)  (3x 2  1)(2 x  5) , find f’(2)
15. Find the equation of the line tangent to the curve
where x = 4.
30
y  2x 1
at the point