MAE C3 Derivertive

Chapter 3
Derivative
MAE101 - CALCULUS
Ly Anh Duong
duongla3@fe.edu.vn
Table of Contents
1 Introduction
▶ Introduction
▶ Derivative
▶ Formulas
▶ Higher-Oder
▶ Chain rule
▶ Implicit
▶ Logarithmic
▶ Applications
▶ Problems
Tangent lines
1 Introduction
• A curve C has equation y = f(x) and we want to find the tangent line to C at
the point P (a, f (a)),
• We consider a nearby point Q(x, f (x)), where x ̸= a
• The slope of the secant line P Q :
f (x) − f (a)
mP Q =
x−a
Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 3 / 50
Tangent lines
1 Introduction
• We let Q approach P along the curve C by letting x approach a,

• If mP Q approaches a number m, then we define the tangent t to be the line
through P with slope m.
f(x) − f(a)
m = lim
x→a x−a
Tangent lines
1 Introduction
DEFINITION The tangent line to the curve y = f (x) at the point P (a, f (a)) is
the line through P with slope
f(x) − f(a)
m = lim
x→a x−a
provided that this limit exists.
If h = x − a, then x = a + h and so the slope of the tangent line in definition
becomes
f (a + h) − f (a)
m = lim
h→0 h
• EXAMPLE I Find an equation of the tangent line to the parabola y = x2 at

the point P (1, 1).
• EXAMPLE 2 Find an equation of the tangent line to the hyperbola y = 3/x
at the point (3, 1).
Table of Contents
2 Derivative
▶ Introduction
▶ Derivative
▶ Formulas
▶ Higher-Oder
▶ Chain rule
▶ Implicit
▶ Logarithmic
▶ Applications
▶ Problems
Definition
2 Derivative
The derivative of a function f at a number a, denoted by f ′ (a), is
f (x) − f (a)
f ′ (a) = lim
x→a x−a
if this limit exists.
or
f (a + h) − f (a)
f ′ (a) = lim
h→0 h
If we replace a in above formular by a variable x, we obtain
f (x + h) − f (x)
f ′ (x) = lim
h→0 h

Definition
2 Derivative
OTHER NOTATIONS
dy df d
f ′ (x) = y ′ = = = f (x) = Df (x) = Dx f (x)
dx dx dx
dy dy

or
dx x=a dx x=a
EXAMPLE
1. Show that a constant function f (x) = k has Derivative f ′ (x) = 0.
2. Find f ′ (x) if f (x) = sinx
 √
1 + 2x − 1
, x ̸= 0

3. Find f ′ (0) if f (x) = x
1 ,x = 0

√
4. Find f ′ (0) if f (x) = 3 x
Differentiable and continuous
2 Derivative
• DEFINITION
A function f is differentiable at a if f ′ (a) exists. It is differentiable on an
open interval (a, b) [or (a, ∞) or (−∞, a) or (−∞, ∞)] if it is differentiable at
every number in the interval.
• THEOREM
If f is differentiable at a, then f continuous at a.

Table of Contents
3 Formulas
▶ Introduction
▶ Derivative
▶ Formulas
▶ Higher-Oder
▶ Chain rule
▶ Implicit
▶ Logarithmic
▶ Applications
▶ Problems
Differentiation formulas
3 Formulas
d d n
1. (k) = 0. 2. (x ) = nxn−1 , n ∈ R.
dx dx
d d
3. sin x = cos x. 4. cos x = − sin x.
dx dx
d d
5. sec x = sec x tan x(secx = 1/cosx). 6. tan x = sec2 x.
dx dx
d d
7. csc x = − csc x cot x(cscx = 1/sinx). 8. cot x = − csc2 x.
dx dx
d x d 1
9. (e ) = ex . 10. (lnx) = .
dx dx x
d x
11. (a ) = ax lna.
dx

Differentiation rules
3 Formulas
Name of rule Function notation

Constant multiple [cf (x)]′ = cf ′ (x)
Sum and difference rule [f (x) ± g(x)]′ = f ′ (x) ± g ′ (x)
Linearity rule [af (x) + bg(x)]′ = af ′ (x) + bg ′ (x)
Product rule [f (x)g(x)]′ = f ′ (x)g(x) + f (x)g ′ (x)
f (x) ′ g(x)f ′ (x) − f (x)g ′ (x)

Quotient rule [g(x) ̸= 0] =
g(x) [g(x)]2
Differentiate the functions given in Problems 1 − 5.
π
1. f (x) = sin x + cos x. 2. g(t) = t2 + cos t + cos .
4
3. f (x) = 2x3 sin x − 3x cos x. 4. p(x) = x2 cos x.
sin x
5. q(x) = .
x
Table of Contents
4 Higher-Oder
▶ Introduction
▶ Derivative
▶ Formulas
▶ Higher-Oder
▶ Chain rule
▶ Implicit
▶ Logarithmic
▶ Applications
▶ Problems
Higher-Order Derivatives
4 Higher-Oder
Leibniz notation
d d
First derivative: y′ f ′ (x) or f (x)
dx dx
d dv d2 y d2

Second derivative y ′′ f ′′ (x) = 2 or f (x)
dx dx ! dx dx2
d d v2 3
d y d3
Third derivative y ′′′ f ′′′ (x) 2
= 3 or f (x)
dx dx dx dx
.. .. ..
. . .
dn y dn
n th derivative y (n) f (n) (x) or f (x)
dxn dxn
In Problems 1 − 4, find f ′ , f ′′ , f ′′′ , and f (4) .
1 1
1. f (x) = x5 − 5x3 + x + 12. 2. f (x) = x8 − x6 − x2 + 2.
4 2

Table of Contents
5 Chain rule
▶ Introduction
▶ Derivative
▶ Formulas
▶ Higher-Oder
▶ Chain rule
▶ Implicit
▶ Logarithmic
▶ Applications
▶ Problems
Chain rule
5 Chain rule
If g is differentiable at x and f is differentiable at g(x), then the composite

function F = f ◦ g defined by F (x) = f (g(x)) is differentiable at x and F′ is given
by the product
F ′ (x) = f ′ (g(x)) · g ′ (x)
In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then
dy dy du
=
dx du dx
Find the Derivative of the function
√
1. y = sin 4x. 2. y = 4 + 3x.
10
3. y = 1 − x2 . 4. y = tan(sin x).

Table of Contents
6 Implicit
▶ Introduction
▶ Derivative
▶ Formulas
▶ Higher-Oder
▶ Chain rule
▶ Implicit
▶ Logarithmic
▶ Applications
▶ Problems
Implicit Differentiation
6 Implicit
√ √
The equation y = 1 − x2 explicitly defines y = f (x) = 1 − x2 as a function of x
for −1 ≤ x ≤ 1,
But the same function can also be defined implicitly by the equation x2 + y 2 = 1,
as long as we restrict y by 0 ≤ y ≤ 1.
To find the derivative of the implicit form:
• Step 1: Differentiate both sides of the equation with respect to x. Remember
that y is really a function of x for part of the curve and use the chain rule.
dy
• Step 2: Solve the differentiated equation algebraically for
dx
EXAMPLE y = f (x) is a differentiable function of x such that
x2 y + 2y 3 = 3x + 2y
dy
find .
dx
Implicit Differentiation
6 Implicit
dy
Find by implicit differentiation in Problems 1-10.
dr
1. x2 + y 2 = 25. 2. x2 + y = x3 + y 3 .
3. xy = 25. 4. xy(2x + 3y) = 2.
1 1 x
5. + = 1. 6. tan = y.
y x y
7. cos xy = 1 − x2 . 8. exy + 1 = x2 .
9. ln(xy) = e2x . 10. exy + ln y 2 = x.

Table of Contents
7 Logarithmic
▶ Introduction
▶ Derivative
▶ Formulas
▶ Higher-Oder
▶ Chain rule
▶ Implicit
▶ Logarithmic
▶ Applications
▶ Problems
Logarithmic Differentiation
7 Logarithmic
Logarithmic differentiation is especially valuable as a means for handling

complicated product or quotient functions and exponential functions where
variables appear in both the base and the exponent.
EXAMPLE
dy
• Find , where y = (x + 1)2x .
dx
e2x (2x − 1)6
• Find the derivative of y = if y > 0
(x3 + 5)2 (4 − 7x)

Table of Contents
8 Applications
▶ Introduction
▶ Derivative
▶ Formulas
▶ Higher-Oder
▶ Chain rule
▶ Implicit
▶ Logarithmic
▶ Applications
▶ Problems
Rate of change
8 Applications
The derivative can be interpreted as a rate of change, which leads to a wide variety
of applications. Viewed as rates of change, derivatives may represent such
quantities as the speed of a moving object, the rate at which a population grows, a
manufacturer’s marginal cost, the rate of inflation, or the rate at which natural
resources are being depleted.
• Average rate of change
CHANGE IN y ∆y f (x + ∆x) − f (x)
= = =
CHANGE IN x ∆x ∆x
∆y f (x + ∆x) − f (x)
• Instaneous rate change = lim = lim = f ′ (x)
∆x→0 ∆x ∆x→0 ∆x
f ′ (x)
• Relative rate of change =
f (x)

Rate of change
8 Applications
EXAMPLE
Let f (x) = x2 − 4x + 7
1. Find the average rate of change of f with respect to x between x = 3 and 5
2. Find the instantaneous rate of change of f at x = 3.

Rectilinear Motion (Modeling in Physics)
8 Applications
ds
An object that moves along a straight line with position s(t) has velocity v(t) =
dt
dv d2 s
and acceleration a(t) = = 2 when these derivatives exist. The speed of the
dt dt
object is |v(t)|.
Example 1
Assume that the position at time t of an object moving along a line is given by
s(t) = 3t3 − 40.5t2 + 162t
for t on [0, 8]. Find the initial position, velocity, and acceleration for the object and
discuss the motion. Compute the total distance traveled.

Rectilinear Motion (Modeling in Physics)
8 Applications
Example 2
A particle moving on the x -axis has position
x(t) = 2t3 + 3t2 − 36t + 40
after an elapsed time of t seconds.

a. Find the velocity of the particle at time t.
b. Find the acceleration at time t.
c. What is the total distance traveled by the particle during the first 3 seconds?

Falling body problems
8 Applications
FORMULA FOR THE HEIGHT OF A PROJECTILE

1
h(t) = − gt2 + v0 t + s0
2
• v0 is the initial velocity.

• s0 is the initial height.
• g is the acceleration due to gravity (32f t/s2 or 9.8m/s2 )

Falling body problems
8 Applications
Example 1
Suppose a person standing at the top of the Tower of Pisa (176ft high ) throws a
ball directly upward with an initial speed of 96ft/s.
a. Find the ball’s height, its velocity, and its acceleration at time t.
b. When does the ball hit the ground and what is its impact velocity?
c. How far does the ball travel during its flight?
Related rates and Applications
8 Applications
SOLVING RELATED RATE PROBLEMS

• Step 1 Draw a figure, if appropriate, and assign variables to the quantities
that vary. Be careful not to label a quantity with a number unless it never
changes in the problem.
• Step 2 Find a formula or equation that relates the variables. Eliminate
unnecessary variables.
• Step 3 Differentiate the equations. You will usually differentiate implicitly
with respect to time.
• Step 4 Substitute specific numerical values and solve algebraically for any
required rate.

8 Applications
Example 1
A person is standing at the end of a pier 12ft above the water and is pulling in a
rope attached to a rowboat at the waterline at the rate of 6ft of rope per minute,
as shown in Figure. How fast is the boat moving in the water when it is 16ft from
the pier?
8 Applications
Example 2
A person 6ft tall is walking away from a street light 20ft high at the rate of 7ft/s.
At what rate is the length of the person’s shadow increasing?
Linear Approximations
8 Applications
We use the tangent line at (a, f (a)) as an approximation to the curve y = f(x)
when x is near a
L(x) = y = f(a) + f’(a)(x - a)

Linear Approximations
8 Applications
The approximation
f (x) ≈ f (a) + f ′ (a)(x − a) = L(x)
is called the linear approximation of f at a.
|x − a| ≤ ϵ, ∀ϵ ≥ 0
√
EX: Determine the linear approximation for√f (x) = 3 x√at x = 8. Use the linear
approximation to approximate the value of 3 8.05 and 3 25.

1 1
f ′ (x) = √
3
, f (8) = 2 f ′ (8) =
3 x2 12
The linear approximation is then,
1 1 4
L(x) = 2 + (x − 8) = x + .
12 12 3
Furthermore, √
3
L(8.05) = 2.00416667 √ 8.05 = 2.00415802
3
L(25) = 3.41666667 25 = 2.92401774
So, at x = 8.05 this linear approximation does a very good job of approximating
the actual value. However, at x = 25 it doesn’t do such a good job.

Differential
8 Applications
Given a function y = f (x) we call dy and dx differentials and the relationship

between them is given by,
dy = f ′ (x)dx
Differentials provide us with a way of estimating the amount a function changes as
a result of a small change in input values.

Differential
8 Applications
EX1: Compute the differential for each of the following.

1. y = t3 − 4t2 + 7t
2. w = x2 sin(2x)

Differential
8 Applications
EX2: The radius of a sphere was measured and found to be 21 cm with a possible
error in measurement of at most 0.05 cm. What is the maximum error in using this
value of the radius to compute the volume of the sphere?

Solution:
4
- If the radius of the sphere is r, then its volume is V = πr3 . - This can be
3
approximated by the differential
dV = 4πr2 dr
= 4π212 0.05 ≈ 277 cm3 .
- The maximum error in the calculated volume is about 277 cm3 .

Table of Contents
9 Problems
▶ Introduction
▶ Derivative
▶ Formulas
▶ Higher-Oder
▶ Chain rule
▶ Implicit
▶ Logarithmic
▶ Applications
▶ Problems
Problems
9 Problems
Prob 1 . Find an equation of the tangent line to the curve at the given point
x−1 2x
a. y = , (3, 2). b. y = , (0, 0).
x−2 x2 + 1
3 − 2x
c. y = 3 − 2x + x2 , x = 1. d. y = , y = −1.
x−1

Problems
9 Problems
Prob 2. Find all the values of x where the tangent line to the graph of the
function is horizontal f (x) = x3 + 4x2 − 11x + 11
11
a. 1. b. ; −1.
3
−11 −11 11
c. ; 1. d. ; ; 1.
3 3 3

Problems
9 Problems
x
Prob 3. Find the derivative of the following function y =
sin x
sin x − x cos x sin x + x cos x
a. . b. .
sin2 x sin2 x
1 sin x − x cos x
c. . d. .
cos x sin x

Problems
9 Problems
Prob 4. Find the derivative of the following function y = xe−2x

a. (1 + 2x)e−2x . b. (1 − 2x)e−2x .
C. e−2x . d. −2e−2x .
2
ex
Prob 5. Find the derivative of the following function y =
sin x
2 2
ex (2x sin x + cos x) ex (2x + cos x)
a. . b. .
2
sin2 x 2
sin2 x
x
e (2x sin x − cos x) x
e (2x sin x − cos x)
C. . d. .
sin2 x sin2 x

Problems
9 Problems
Prob 6. Find y ′′
a. y = xe 3x−1
√3
b. y = 2x + 1
C. y = e−x cos x
Prob 7. The position in feet of race car along a straight track after t seconds is
1
modeled by the function s(t) = 8t2 − t3
16
a. Find the average velocity of the vehicle over the time interval [4, 4.1]
b. Find the instantaneous velocity of the vehicle at t = 4 seconds.

Problems
9 Problems
Prob 8. A toy company can sell x electronic gaming systems at a price of p =
−0.01x + 400 dollars per gaming system. The cost of manufacturing x systems is
given by C(x) = 100x + 10000 dollars. Find the rate of change of profit when
10000 game produced. Should the toy company increase or decrease
production?(profit=x.p-c(x))

Problems
9 Problems
Prob 9. A table of values for f, f ′ , g and g ′ is given

x f (x) g(x) f ′ (x) g ′ (x)
1 3 2 4 6
2 1 8 5 7
3 7 2 7 9
a. If h(x) = f (g(x)), find h′ (1)
b. If H(x) = g(f (x)), find H ′ (1)
c. If F (x) = f (f (x)), find F ′ (2)
d. If G(x) = g(g(x)),pfind G′ (3).
Prob 10. If h(x) = 4 + 3f (x), where f (1) = 7, f ′ (1) = 4, find h′ (1).
Prob 11. Find f ′ in terms of g ′
a. f (x) = g(sin 2x)
b. f (x) = g e1−3x .
Problems
9 Problems
dy
Prob 12. Find for:
dt
dx
a. y = x3 + x + 2, = 2 and x = 1.
dt
dx
b. y = ln x, = 1 and x = e2
dt ′
Prob 13. Find y by implicit differentiation
a. x4 + y 4 = 16x + y
b. x3 + xy = y 2
Prob 14. Find the linearization L(x) of the function at a
1
a. f (x) = √ , a=2
√2 +x
b. f (x) = 3 5 − x, a = −3.
Prob 15. Each side of a square is increasing at a rate of 7 cm/s. At what rate is
the area of the square increasing when the area of the square is 25 cm2 ?
Problems
9 Problems
2
Prob 16. Compute the differential of the function y = e2x
2
a. dy = 4xe2x dx
2
b. dy = 4e2x dx
2
c. dy = 2xe2x dx
2
d. dy = e2x dx
Prob 17. Compute the differential of the function y = ln(sin x)
a. dy = cot xdx
b. dy = sin xdx
c. dy = tan xdx
cos x ln(sin x)dx

Problems
9 Problems
x−1
Prob 18. Let y = . Compute the value of dy(1)
x+1
a. dy(1) = 2dx
1
b. dy(1) = dx
4
1
c. dy(1) = dx
2
d. dy(1) = dx
ex
Prob 19. Let y = x . Compute dy(ln 2)
e +1
1
a. dy(ln 2) = dx
2
4
b. dy(ln 2) = dx
9
2
c. dy(ln 2) = dx
9
Q&A
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Chapter 3

• We let Q approach P along the curve C by letting x approach a,

• EXAMPLE I Find an equation of the tangent line to the parabola y = x2 at

The derivative of a function f at a number a, denoted by f ′ (a), is

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 7 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 9 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 11 / 50

Name of rule Function notation

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 14 / 50

If g is differentiable at x and f is differentiable at g(x), then the composite

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 16 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 19 / 50

Logarithmic differentiation is especially valuable as a means for handling

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 21 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 23 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 24 / 50

s(t) = 3t3 − 40.5t2 + 162t

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 25 / 50

x(t) = 2t3 + 3t2 − 36t + 40

after an elapsed time of t seconds.

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 26 / 50

FORMULA FOR THE HEIGHT OF A PROJECTILE

• v0 is the initial velocity.

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 27 / 50

SOLVING RELATED RATE PROBLEMS

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 29 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 32 / 50

f (x) ≈ f (a) + f ′ (a)(x − a) = L(x)

is called the linear approximation of f at a.

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 33 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 34 / 50

Given a function y = f (x) we call dy and dx differentials and the relationship

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 35 / 50

EX1: Compute the differential for each of the following.

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 36 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 37 / 50

- The maximum error in the calculated volume is about 277 cm3 .

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 38 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 40 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 41 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 42 / 50

Prob 4. Find the derivative of the following function y = xe−2x

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 43 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 44 / 50

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 45 / 50

Prob 9. A table of values for f, f ′ , g and g ′ is given

Ly Anh Duong Chapter 3 Derivative duongla3@fe.edu.vn 48 / 50

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