OPERATIONS WITH DERIVATIVES V 0.3
For this part, it will be assumed the derivatives of the basic functions
f HxL
df HxL
dx
c, constant
mx + b
0
m
xn , n ¹ 0
sin HxL
cos HxL
nxn-1
cos HxL
-sin HxL
The derivative of a constant times a function
The derivative of a constant multiplying a function is the constant multiplied by the derivative of the function. It
is represented as
Ha f HxLL¢ = a f ¢ HxL
or
d
@a
dx
f HxLD = a
d
dx
f HxL
This result can be justified using the limit definition
Ha f HxLL ¢ =Lim
h®0
a f Hx+hL-a f HxL
h
= a Lim
h®0
f Hx+hL- f HxL
h
= a f ¢ HxL
EXAMPLE 1
i.
ii.
d
dx
d
dx
I3 x3 M = 3
J
d
dx
Ix3 M = 9 x2
-2
d
N = -2 d x
x
1
J x N = -2
d
dx
Ix-1 M = 2 x-2 =
2
x2
EXERCISE 1
Apply the rule above to calculate the derivative of each of the following functions. Then use the that result to find the
equation of the tangent line at the indicated point.
a. y = 2 x4
2
3
b. y =
a = -2
x
a=4
c. y = 3 sinHxL
d. y =
e. y =
3
2 x3
a = -1
-3
a = 27
3
x
a=Π
2
Derivatives Operations with Derivativesv03.nb
Apply the rule above to calculate the derivative of each of the following functions. Then use the that result to find the
equation of the tangent line at the indicated point.
a. y = 2 x4
2
3
b. y =
a = -2
x
a=4
c. y = 3 sinHxL
d. y =
e. y =
a=Π
3
2 x3
a = -1
-3
a = 27
3
x
Finding the critical points of a function y = f HxL defined by a formula
Remember that the function has to be continuous on the interval in consideration. This is a 2-step process.
STEP 1. Find all the points where f ¢ HxL = 0 or where f ¢ HxL is undefined. These are the candidates to be critical
points.
STEP 2. From the candidate points choose the ones in the domain of the function f HxL. Those are the critical points
EXAMPLE 2
To find the critical points of the function y =
x we use y¢ HxL =
1
2
x
. Remember that a critical point has to be in the
domain of the function, which in this case is @0, ¥L
i. Need to find the points in the domain where y¢ HxL = 0 or y¢ HxL is undefined.
y¢ HxL = 0 –
1
2
x
= 0 This equation has no solution. So there are not critical points where the derivative is
zero
y¢ HxL is undefined at x = 0. But we know that automatically x=0 is a critical point since this is an end point in the
domain.
ii. So, the only critical point is when x = 0.
EXERCISE 2
Algebraically, find the critical points for each of the following functions
a. y = 2 x - 3
b. y = x3
c. y =
3
d. y =
4
x
x
e. y = 3 x3 on the interval @1, 5D
f. y =
4
x
4
Derivatives Operations with Derivativesv03.nb
EXERCISE 2
Algebraically, find the critical points for each of the following functions
a. y = 2 x - 3
b. y = x3
c. y =
3
d. y =
4
x
x
e. y = 3 x3 on the interval @1, 5D
f. y =
4
x
g. y =
4
x
on the interval [1,10]
Additive Property of Derivatives
The derivative of the sum (or difference) of two functions is the sum (or difference) of the derivatives in their common
domain. It can be summarized by saying that if f HxL and gHxL are differentiable functions on a common domain then
H f HxL ± gHxLL' = H f ± gL ¢ HxL = f ¢ HxL ± g ¢ HxL
d
dx
H f HxL ± gHxLL =
d
dx
f HxL ±
This is the justification of this result using limits.
H f +gL HxL
H f + gL¢ HxL = LimH f + gL Hx + hL -
h
h®0
= Lim
f Hx+hL+gHx+hL- f HxL-gHxL
h
h®0
= LimJ
f Hx+hL- f HxL
h
h®0
= Lim
f Hx+hL- f HxL
h®0
¢
h
gHx+hL-gHhL
+
h
N
gHx+hL-gHhL
+ Lim
h
h®0
= f HxL + g ¢ HxL
EXAMPLE 3
i. For y = 3 + x2 +
x , y ¢ HxL = 0 + 2 x +
1
2
x
ii. For f HxL = sinHxL - 3 x + 2, f ¢ HxL = cosHxL - 3
The next two examples combine properties the two cases already discussed.
iii. For y =
1
2
1
'
x4 + 3 x - 5, y ¢ HxL = I 2 x4 M + H3 xL' - H5L'
= 2 x3 + 3
iv.
d
dt
2 t3 -
t +
2
3
3
t2 =
d
dt
I2 t3 M -
d
d t
= 6 t2 = 6 t2 -
It12 M +
1 -12
t
2
1
2
+
9
2
J 3 t23 N
4 -13
t
9
4
+
t
d
dt
3
t
d
dx
gHxL
3
4
Derivatives Operations with Derivativesv03.nb
The next two examples combine properties the two cases already discussed.
iii. For y =
1
2
1
'
x4 + 3 x - 5, y ¢ HxL = I 2 x4 M + H3 xL' - H5L'
= 2 x3 + 3
iv.
d
dt
2 t3 -
2
3
t +
3
t2 =
d
dt
I2 t3 M -
d
d t
= 6 t2 = 6 t2 -
It12 M +
1 -12
t
2
1
2
+
2
J 3 t23 N
4 -13
t
9
4
+
t
d
dt
9
3
t
EXERCISE 3
Find the derivative for each of the following functions.
a. y = x2 - 2 x
b. y = 4 x5 -
3
x
+þ
c. y = 2 sinHxL +
d. y = 5 +
2
3
+
x
x
x
3
+
2
5
EXERCISE 4
For each of the functions
I. y = x + x
II. f HxL = x2 -
2
x
III. gHzL = 5 - x +
2
x2
a. Find the domain and points of discontinuity.
b. Find the derivative function algebraically.
c. Find its critical points and classify them as points where the derivative is zero or undefined.
Derivative of the product of two functions
The derivative of the product of two functions is
@ f HxL gHxLD' = f ' HxL gHxL + f HxL g ' HxL
d
dx
H f HxL gHxLL = gHxL
d
dx
f HxL + f HxL
d
dx
gHxL
EXERCISE 4
Each of the following functions can be represented as the product of functions f and g. Clearly identify them and use the the
product rule for derivatives to find its derivative.
a. y = x2 sinHxL
b. y =
1
x J2 x - x N
Derivatives Operations with Derivativesv03.nb
EXERCISE 4
Each of the following functions can be represented as the product of functions f and g. Clearly identify them and use the the
product rule for derivatives to find its derivative.
a. y = x2 sinHxL
b. y =
1
x J2 x - x N
c. y = x LnHxL
d. y = x ã x
e. y = cosHxL * I3 x4 + x -
f. y = I2 x5 +
1
2
x - 3M
3
xM
x2 +
2
x
Derivative of the quotient of two functions
The derivative of the quotient of two functions is
f HxL
B gHxL F' =
d
dx
J
f HxL
gHxL
f ' HxL‰gHxL- f HxL‰g' HxL
@gHxLD2
gHxL
N=
d
dx
f HxL- f HxL
d
dx
gHxL
@gHxLD2
EXERCISE 5
Each of the following functions can be represented as the quotient of two functions f and g. Clearly identify them and use the
the quotient rule for derivatives to find its derivative. Some of these derivatives can be calculated using previous rules as well.
When it is possible you are asked to do it.
a. y =
x+1
x-1
b. y =
x
x2 -2 x
c. y =
x5
120
(two ways)
5
6
Derivatives Operations with Derivativesv03.nb
EXERCISE 5
Each of the following functions can be represented as the quotient of two functions f and g. Clearly identify them and use the
the quotient rule for derivatives to find its derivative. Some of these derivatives can be calculated using previous rules as well.
When it is possible you are asked to do it.
a. y =
x+1
x-1
b. y =
x
x2 -2 x
c. y =
x5
120
d. y =
x3 -2 x
x2
e. y =
2x -x2
sinHxL
f. y =
4
x3
(two ways)
(two ways)
(two ways)
APPLICATIONS
I. The number of cases of AIDS in the USA, when it appeared, grew approximately according to the model
AHtL = 175 t3 , where t is the number of year since 1972 when the epidemic appeared. Use this model to
a. Determine the number of cases of AIDS in 1980 using this model.
b. How fast was the number of people with AIDS increasing in 1980?
c. From the information from (a)-(b) predict the number of cases in 2000 under this model. Is this an
overestimate? Underestimate according to the model? What characteristic of the graph let you determine whether
it is an under/over estimate?
d. Use integration to determine the average number of cases of AIDS in the USA during the period
1972-200
II. The total mass of a population is the product of the number of individuals and the mass of each individual.
Certain population is given by the formula PHtL = 2.0‰ 106 + 2.0‰ 104 t , and the mass per person is modeled by
W HtL = 80 - 0.005 t2 , where t is measured in years and mass is measured in kilograms.
a. Find a formula for the mass of the population after t years. Call it M = M HtL
b. Find the expression for M ' HtL
c. Find all the critical points for the function M = M HtL
d. Calculate MH10L and M' H10L and use those values to estimate MH13L
Derivatives Operations with Derivativesv03.nb
II. The total mass of a population is the product of the number of individuals and the mass of each individual.
Certain population is given by the formula PHtL = 2.0‰ 106 + 2.0‰ 104 t , and the mass per person is modeled by
W HtL = 80 - 0.005 t2 , where t is measured in years and mass is measured in kilograms.
a. Find a formula for the mass of the population after t years. Call it M = M HtL
b. Find the expression for M ' HtL
c. Find all the critical points for the function M = M HtL
d. Calculate MH10L and M' H10L and use those values to estimate MH13L
III. (Bird Reproduction) A certain bird population has the following model for offspring reproduction. When a
bird lays N eggs, the probability that each chick survives is PHNL = 1 - 0.1 N -notice that P(N) is always a
number between 0 and 1-.
a. What is the probability to survive when N=1, 4, 7, 10. Give the answer as a percentage. Write a
complete sentence in each case.
b. Mathematically the total number of off springs that are likely to survive is given by SHNL = N ‰ PHNL.
Find a formula for S(N)
c. Justify why S(N) is defined only on the interval [0,10]
d. Find the equation of the tangent line to the S(N) when N=3. Use the equation to estimate when S(N)=0.
d. Using the rules of derivatives, find all the critical points of S=S(N). Use the graph of S=S(N) to explain
what happens at each of the critical points.
7