Ch 3 the derivative

CALCULUS I
Lecturer: Eng Fuad Abdirizak Elmi

Department of Civil Engineering
University Of Somalia
CHAPTER 3
THE DERIVATIVE

Introduction
 Many real-world phenomena involve changing quantities —
the speed of a rocket, the inflation of currency, the number of
bacteria in a culture, the shock intensity of an earthquake, the
voltage of an electrical signal, and so forth.
 In this chapter we will develop the concept of a “derivative,”
which is the mathematical tool for studying the rate at which
one quantity changes relative to another.
 The study of rates of change is closely related to the geometric
concept of a tangent line to a curve, so we will also be
discussing the general deﬁnition of a tangent line and methods
for ﬁnding its slope and equation.

 Outline
Here is a listing and brief description of the material in
this set of notes.
• Tangent lines and rates of change – In this section we define
the derivative, give various notations for the derivative and
work a few problems
• The Derivative of Functions – In this section we give most of
the general derivative formulas and properties used when
taking the derivative of a function.
• The rules of derivatives – In this section we will give
important formulas for differentiating functions.

 Outline
• Derivative of Trigonometry functions – In this section we will
discuss differentiating trig functions.
• The Chain Rule – In this section we discuss one of the more
useful and important differentiation formulas, The Chain Rule.
• Implicit Differentiation– In this section we will discuss
implicit differentiation. Not every function can be explicitly
written in terms of the independent variable.

3.1 Tangent lines and rates of change
Introduction
 In this section we will discuss three ideas: tangent lines to
curves, the velocity of an object moving along a line, and the
rate at which one variable changes relative to another.
 Our goal is to show how these seemingly unrelated ideas are,
in actuality, closely linked.

Tangent Line
 We showed how the notion of a limit could be used to ﬁnd an
equation of a tangent line to a curve.
 At that stage in the text we did not have precise deﬁnitions of
tangent lines and limits to work with, so the argument was
intuitive and informal.

Tangent Line
 Suppose that xo is in the domain of the function f. The tangent
line to the curve y =f(x) at the point P(xo, f(xo)) is the line with
equation
𝑦 − 𝑓 𝑥 𝑜 = 𝑚 𝑡𝑎𝑛(𝑥 − 𝑥 𝑜)
 Where
𝑚 𝑡𝑎𝑛 = lim
𝑥→𝑥 𝑜
𝑓 𝑥 − 𝑓 𝑥 𝑜
𝑥 − 𝑥 𝑜

Tangent Line
 There is an alternative way of expressing equation (1) that is
commonly used. If we let h denote the difference
ℎ = 𝑥 − 𝑥 𝑜
 then the statement that x→x0 is equivalent to the statement
h→0, so we can rewrite (1) in terms of xo and h as
𝑚 𝑡𝑎𝑛 = lim
ℎ→0
𝑓 𝑥 𝑜 + ℎ − 𝑓 𝑥
ℎ

Velocity
 One of the important themes in calculus is the study of motion.
 To describe the motion of an object completely, one must
specify its speed (how fast it is going) and the direction in
which it is moving.
 The speed and the direction of motion together comprise what
is called the velocity of the object.

Velocity
 For example, knowing that the speed of an aircraft is 500 mi/h
tells us how fast it is going, but not which way it is moving.
 In contrast, knowing that the velocity of the aircraft is 500
mi/h due south pins down the speed and the direction of
motion.

Slopes and Rate of change
 Velocity can be viewed as rate of change — the rate of change
of position with respect to time. Rates of change occur in other
applications as well. For example:
 A microbiologist might be interested in the rate at which the
number of bacteria in a colony changes with time.
 An engineer might be interested in the rate at which the length
of a metal rod changes with temperature.
 A medical researcher might be interested in the rate at which
the radius of an artery changes with the concentration of
alcohol in the bloodstream.

3.2 The Derivative of Function
Introduction
 In this section we will discuss the concept of a “derivative,”
which is the primary mathematical tool that is used to calculate
and study rates of change.

Introduction
 In the last section we showed that if the limit exists
lim
ℎ→0
ℎ
 Then it can be interpreted either as the slope of the tangent line
to the curve y =f(x) at x =x0.

Definition
 The function 𝑓′ deﬁned by the formula
𝑓′ 𝑥 = lim
ℎ→0
𝑓 𝑥 + ℎ − 𝑓 𝑥
ℎ
 is called the derivative of f with respect to x.
 The domain of 𝑓 consists of all 𝑥 in the domain of 𝑓 for which
the limit exists.
 The term “derivative” is used because the function 𝑓 is derived
from the function 𝑓 by a limiting process.

Examples
 Find the derivative with respect to x of
i. 𝑓 𝑥 = 𝑥
ii. 𝑓 𝑥 = 𝑥2
iii. 𝑓 𝑥 = 𝑥3
iv. 𝑓 𝑥 = 16𝑥 + 35
v. 𝑓 𝑥 = 7𝑥2+10x

3.3 The Rules of Derivative
Introduction
 In the last section we deﬁned the derivative of a function f as a
limit, and we used that limit to calculate a few simple
derivatives.
 In this section we will develop some important theorems that
will enable us to calculate derivatives more efﬁciently.

Notations
 f x “f prime x” or “the derivative of f with respect to x”
y “y prime”
dy
dx
“dee why dee ecks” or “the derivative of y with respect
to x”
df
dx
“dee eff dee ecks” or “the derivative of f with respect to
x”
 
d
f x
dx
“dee dee ecks uv eff uv ecks” or “the derivative of f of x”


Derivative of a Constant
 The simplest kind of function is a constant function 𝑓 𝑥 = 𝑐.
 Since the graph of 𝑓 is a horizontal line of slope 0, the tangent
line to the graph of 𝑓 has slope 0 for every x and hence we can
see geometrically that 𝑓 𝑥 = 0 (Figure 2.3.1).

 We can also see this algebraically since
𝑓′ 𝑥 = lim
ℎ→0
ℎ
= lim
ℎ→0
𝑐 − 𝑐
ℎ
= lim
ℎ→0
= 0
 The derivative of a constant function is 0; that is, if c is any
real number, then
𝑑
𝑑𝑥
𝑐 = 0

 Examples
i.
𝑑
𝑑𝑥
1 = 0
ii.
𝑑
𝑑𝑥
−3 = 0
iii.
𝑑
𝑑𝑥
π = 0
iv.
𝑑
𝑑𝑥
− 2 = 0
v.
𝑑
𝑑𝑥
100 = 0

Derivative of a Power Function
 The simplest power function is 𝑓 𝑥 = 𝑐. Since the graph of
𝑓 is a line of slope 1, 𝑓′
𝑥 = 1 or in other words
𝑑
𝑑𝑥
𝑥 = 1
 (The Power Rule) If n is an integer, then
𝑑
𝑑𝑥
𝑥 𝑛
= 𝑛𝑥 𝑛−1

Derivative of a Power Function
 Examples
i.
𝑑
𝑑𝑥
𝑥4 = 4𝑥3
ii.
𝑑
𝑑𝑥
𝑥5
= 5𝑥4
iii.
𝑑
𝑑𝑡
𝑡12 = 12𝑡11
iv.
𝑑
𝑑𝑥
𝑥7
= 7𝑥6
v.
𝑑
𝑑𝑡
𝑡−10 = −10𝑥−11

Derivative of a Constant Times a Function
 (Constant Multiple Rule) If 𝑓 is differentiable at 𝑥 and 𝑐 is any
real number, then c𝑓 is also differentiable at 𝑥 and
𝑑
𝑑𝑥
𝑐𝑓(𝑥) = 𝑐
𝑑
𝑑𝑥
𝑓(𝑥)
𝑐𝑓 ′ = 𝑐𝑓′

Derivative of a Constant Times a Function
 Examples
i.
𝑑
𝑑𝑥
4𝑥8
ii.
𝑑
𝑑𝑥
−2𝑥12
iii.
𝑑
𝑑𝑥
5𝑥4
iv.
𝑑
𝑑𝑥
4𝑥−4
v.
𝑑𝑦
𝑑𝑥
3
𝑥4

Derivative of Sums and Differences
 (Sum and Difference Rules) If f and g are differentiable at x,
then so are 𝑓 + 𝑔 and 𝑓 − 𝑔 and
 In words, the derivative of a sum equals the sum of the
derivatives, and the derivative of a difference equals the
difference of the derivatives.
𝑑
𝑑𝑥
𝑓 𝑥 ± 𝑔(𝑥) =
𝑑
𝑑𝑥
𝑓(𝑥) ±
𝑑
𝑑𝑥
𝑔(𝑥)
𝑓 ± 𝑔 ′
= 𝑓′
± 𝑔′

Derivative of Sums and Differences
 Examples
i. 𝑦 = 2𝑥6 + 𝑥−4
ii. 𝑓 𝑥 = 𝑥3
− 3𝑥2
+ 4
iii. 𝑓 𝑥 =
8
𝑥3 − 2𝑥4 + 6𝑥2 − 4
iv. 𝑓 𝑥 = 3𝑥8 − 2𝑥2 + 4𝑥 + 1
v. 𝑓 𝑥 = 𝑥5
+ 5𝑥2
& 𝑔 𝑥 = 43
− 7𝑥  𝑓 ± 𝑔 ′

Higher Derivatives
 The derivative 𝑓′ of a function 𝑓 is itself a function and hence
may have a derivative of its own.
 If 𝑓′ is differentiable, then its derivative is denoted by 𝑓′′and
is called the second derivative of 𝑓.
𝑓, 𝑓′′ = 𝑓′ ′, 𝑓′′′ = 𝑓′′ ′, 𝑓′′′ = 𝑓′′ ′

Higher Derivatives
 Examples
i. 𝑓 𝑥 = 3𝑥4 − 2𝑥3 + 𝑥2 − 4𝑥 + 2
ii. 𝑓 𝑥 = 10𝑥4
+ 6𝑥3
− 5𝑥2
+ 20𝑥 − 4
iii. 𝑓 𝑥 = 2𝑥5 + 4𝑥4 − 10𝑥3 + 8𝑥3 + 𝑥 − 5

Derivative a Product
• (The Product Rule) If 𝑓 and 𝑔 are differentiable at 𝑥, then so is
the product 𝑓 ∙ 𝑔, and
• In words, the derivative of a product of two functions is the
ﬁrst function times the derivative of the second plus the second
function times the derivative of the ﬁrst.
𝑑
𝑑𝑥
𝑓 𝑥 ∙ 𝑔(𝑥) = 𝑓 𝑥
𝑑
𝑑𝑥
𝑔(𝑥) +
𝑑
𝑑𝑥
𝑓(𝑥) 𝑔(𝑥)
𝑓 ∙ 𝑔 ′ = 𝑓 ∙ 𝑔′ + 𝑓′ ∙ 𝑔

Derivative a Product
 Examples
i. 4𝑥2 − 1 7𝑥3 + 𝑥  𝑓 ∙ 𝑔 ′
ii. 3𝑥3
− 3𝑥 2𝑥 − 10  𝑓 ∙ 𝑔 ′
iii. 𝑥2 − 2 𝑥3 − 5  𝑓 ∙ 𝑔 ′

Classwork
 Find the derivatives of the given functions
i. 𝑓 𝑥 = 𝑥
ii.
𝑑
𝑑𝑥
=
𝜋
𝑥
iii. 𝑦 = 5𝑥3 + 12𝑥2 − 15
iv. 𝑦 = −6𝑥5 + 3𝑥2 − 51𝑥
v. 𝑓 𝑥 = (𝑥2
−3𝑥 + 2) & 𝑔 𝑥 = (2𝑥3
− 5𝑥)  𝑓 ± 𝑔 ′

Classwork
vi. 𝑦 = 3𝑥4
+ 2𝑥7
− 7𝑥−3
+ 4𝑥 𝑓′′
′
vii. 𝑦 = 4𝑥5
− 2𝑥4
+ 9𝑥3
+ 10𝑥−2
− 13𝑥 𝑓′′
′
viii. 𝑦 = 𝑥2 + 3𝑥 2
ix. 3𝑥2 + 6 2𝑥 − 4  𝑓 ∙ 𝑔 ′

Homework
i. 𝑦 = 3𝑥2 − 𝑥3 + 15𝑥2 − 𝑥
ii. 𝑦 = 𝑥3
+ 2𝑥2
+ 4𝑥
iii. 𝑓 𝑥 = (𝑥3−3𝑥2) & 𝑔 𝑥 = (4𝑥3 − 12𝑥2 + 2)
 𝑓 ± 𝑔 ′
iv. 𝑦 = 2𝑥4 + 4𝑥3 − 30𝑥2 + 3𝑥 + 4 𝑓′′′
v. 3𝑥 − 5 2𝑥 + 3  𝑓 ∙ 𝑔 ′

Homework
vi. 𝑥 − 2 2 𝑥2 + 2𝑥 + 4  𝑓 ∙ 𝑔 ′
vii. 4𝑥2
+ 3𝑥 2
10𝑥2
+ 4𝑥  𝑓 ∙ 𝑔 ′

Derivative a Quotient
• Just as the derivative of a product is not generally the product
of the derivatives,
• so the derivative of a quotient is not generally the quotient of
the derivatives.
• The correct relationship is given by the following theorem.

• (The Quotient Rule) If 𝑓 and 𝑔 are both differentiable at 𝑥 and
if 𝑔(𝑥) ≠ 0, then 𝑓
𝑔 is differentiable at 𝑥 and
𝑑
𝑑𝑥
𝑓 𝑥
𝑔(𝑥)
=
𝑔 𝑥
𝑑
𝑑𝑥
𝑓 𝑥 − 𝑓 𝑥
𝑑
𝑑𝑥
𝑔(𝑥)
𝑔(𝑥) 2
𝑓
𝑔
′
=
𝑓′ ∙ 𝑔 − 𝑓 ∙ 𝑔′
𝑔2

 Examples
i. 𝑓 𝑥 =
𝑥3+2𝑥2−1
𝑥+5
ii. 𝑓 𝑥 =
𝑥2−1
𝑥4+1
iii. 𝑓 𝑥 =
3𝑥2
𝑥2+2
iv. 𝑓 𝑥 =
4𝑥+4
𝑥−3

 Examples
v. 𝑓 𝑥 =
2𝑥2+5
3𝑥−4
vi. 𝑓 𝑥 =
𝑥2+𝑥−2
𝑥3+6
vii. 𝑓 𝑥 =
𝑥+4 2
5𝑥3+12𝑥2

Summary Of Differentiation Rules

Classwork
i. 𝑓 𝑥 =
𝑥2−1
𝑥2+2
ii. 𝑓 𝑥 =
𝑥3−10
2𝑥2
iii. 𝑓 𝑥 =
3𝑥+4
𝑥2+1
iv. 𝑓 𝑥 =
2𝑥2+4𝑥−6
𝑥2+3𝑥

3.4 The Derivative of Trigonometric Functions
Introduction
• The main objective of this section is to obtain formulas for the
derivatives of the six basic trigonometric functions.
• We will assume in this section that the variable x in the
trigonometric functions sin x, cos x, tan x, cot x ,sec x, and
csc x is measured in radians

Introduction
• The derivatives of the six basic trigonometric functions are:-

Examples
i.
𝑑
𝑑𝑥
5 sin 𝑥 + 4 tan 𝑥
ii.
𝑑
𝑑𝑥
𝑥2 + sin 𝑥 + 4 cos 𝑥
iii.
𝑑
𝑑𝑥
8 s𝑒𝑐 𝑥 − 5 cos 𝑥
iv.
𝑑
𝑑𝑥
𝑥 sin 𝑥
v.
𝑑
𝑑𝑥
𝑥2 sin 𝑥

Examples
vi.
𝑑
𝑑𝑥
𝑥2 sin 𝑥 tan 𝑥
vii.
𝑑
𝑑𝑥
𝑥3
𝑐𝑜𝑠𝑥
viii.
𝑑
𝑑𝑥
sin 𝑥
cos 𝑥
ix.
𝑑
𝑑𝑥
sin 𝑥
1+cos 𝑥

Classwork
i.
𝑑
𝑑𝑥
2 cot 𝑥 − 7 csc 𝑥
ii.
𝑑
𝑑𝑥
5 sin 𝑥 cos 𝑥 + 4 csc 𝑥
iii.
𝑑
𝑑𝑥
3 s𝑒𝑐 𝑥 − 10 co𝑡 𝑥
iv.
𝑑
𝑑𝑥
sin 𝑥
3−2co𝑡 𝑥

3.5 Derivative Logarithmic functions
Introduction
 In this section we will obtain derivative formulas for
exponential functions.
 we will explain why the natural logarithm function is preferred
over logarithms with other bases in calculus.

Introduction
 We will establish that 𝑓 𝑥 = ln 𝑥 is differentiable for 𝑥 > 0
by applying the derivative deﬁnition to 𝑓(𝑥).
 To evaluate the resulting limit, we will need the fact that ln 𝑥 is
continuous for 𝑥 > 0 (Theorem 1.6.3), and we will need the
limit
𝑑
𝑑𝑥
𝑙𝑛 𝑢 =
𝑢′
𝑢

Examples
i.
𝑑
𝑑𝑥
𝑙𝑛 𝑥
ii.
𝑑
𝑑𝑥
𝑙𝑛 𝑥2
iii.
𝑑
𝑑𝑥
𝑙𝑛 3𝑥3
iv.
𝑑
𝑑𝑥
𝑙𝑛 𝑥 + 5
v.
𝑑
𝑑𝑥
𝑙𝑛(𝑠𝑖𝑛𝑥)
vi.
𝑑
𝑑𝑥
𝑙𝑛 7𝑥 + 5 − 𝑥3

Classwork
i.
𝑑
𝑑𝑥
𝑙𝑛 𝑥3
ii.
𝑑
𝑑𝑥
𝑙𝑛 4𝑥4
+ 6𝑥2
iii.
𝑑
𝑑𝑥
𝑙𝑛 6𝑥
iv.
𝑑
𝑑𝑥
𝑙𝑛 3𝑥 + 4
v.
𝑑
𝑑𝑥
𝑙𝑛 3𝑥5 + 5𝑥−2 + 4𝑥

3.6 Derivative of Exponential
Introduction
 In this section we will obtain derivative formulas for
exponential functions.
 To differentiate exponential function, Let u is a differentiable
function of x, then the equation is
𝑑
𝑑𝑥
𝑒 𝑢 = 𝑒 𝑢 𝑢′

Examples
i.
𝑑
𝑑𝑥
𝑒 𝑥
ii.
𝑑
𝑑𝑥
𝑒5𝑥+3
iii.
𝑑
𝑑𝑥
𝑒 𝑥3+8𝑥
iv.
𝑑
𝑑𝑥
𝑒sin 𝑥
v.
𝑑
𝑑𝑥
𝑥3 𝑒4𝑥

Classwork
i.
𝑑
𝑑𝑥
𝑒 𝑥2
ii.
𝑑
𝑑𝑥
3𝑥2
𝑒4𝑥2
iii.
𝑑
𝑑𝑥
𝑒3𝑥3+6𝑥
iv.
𝑑
𝑑𝑥
𝑒cos 𝑥
v.
𝑑
𝑑𝑥
tan 𝑥 𝑒 𝑥2

3.7 The Chain rule
Introduction
 In this section we will derive a formula that expresses the
derivative of a composition 𝑓°𝑔 in terms of the derivatives of
𝑓 and 𝑔.
 This formula will enable us to differentiate complicated
functions using known derivatives of simpler functions.

3.7 The Chain rule
Generalized Derivative Formulas
 We can rewrite the generalized derivative formula by using the
derivative of 𝑓(𝑥) to produce the derivative of 𝑓(𝑢) , where u
is a function of x.
𝑑
𝑑𝑥
𝑓 𝑔 𝑥 = 𝑓′
𝑔 𝑥 𝑔′
𝑑
𝑑𝑥
𝑈 𝑛 = 𝑛 𝑈 𝑛−1 𝑈′

3.7 The Chain rule
Examples
i.
𝑑
𝑑𝑥
5𝑥 + 8 4
ii.
𝑑
𝑑𝑥
10𝑥2 − 3 3
iii.
𝑑
𝑑𝑥
4𝑥3 + 3𝑥2 − 5𝑥 6
iv.
𝑑
𝑑𝑥
sin(6𝑥)
v.
𝑑
𝑑𝑥
cos 𝑥2
vi.
𝑑
𝑑𝑥
𝑙𝑛 𝑥 7

3.7 The Chain rule
Examples
vii.
𝑑
𝑑𝑥
1
𝑥2+8𝑥 3
viii.
𝑑
𝑑𝑥
2𝑥−3
4+5𝑥
4

3.7 The Chain rule
Classwork
i.
𝑑
𝑑𝑥
6𝑥4 + 7𝑥2 + 3 3
ii.
𝑑
𝑑𝑥
sec (4𝑥)
iii.
𝑑
𝑑𝑥
tan 𝑥3
iv.
𝑑
𝑑𝑥
𝑙𝑛 𝑥3 3
v.
𝑑
𝑑𝑥
𝑥3 − 7

3.7 The Chain rule
Classwork
vi.
𝑑
𝑑𝑥
6 𝑥3 + 2𝑥 100

Homework
i.
𝑑
𝑑𝑥
𝑙𝑛 4𝑥3 + 15𝑥
ii.
𝑑
𝑑𝑥
𝑙𝑛 100𝑥
iii.
𝑑
𝑑𝑥
𝑙𝑛 10𝑥10 + 15𝑥2 + 4
iv.
𝑑
𝑑𝑥
𝑙𝑛 sec 𝑥

Homework
i.
𝑑
𝑑𝑥
𝑒50𝑥2
ii.
𝑑
𝑑𝑥
10𝑥3
𝑒4𝑥2
iii.
𝑑
𝑑𝑥
𝑒4𝑥2+2𝑥−6
iv.
𝑑
𝑑𝑥
𝑒ta𝑛 𝑥

3.7 The Chain rule
Homework
i.
𝑑
𝑑𝑥
20𝑥−4 + 3𝑥2 + 8𝑥 6
ii.
𝑑
𝑑𝑥
csc (4𝑥)
iii.
𝑑
𝑑𝑥
tan(𝑥3 + 4)
iv.
𝑑
𝑑𝑥
𝑥3 − csc 𝑥

3.8 The Implicit Differentiation
Introduction
 Up to now we have been concerned with differentiating
functions that are given by equations of the form y = (𝑓 𝑥 ).
 In this section we will consider methods for differentiating
functions for which it is inconvenient or impossible to express
them in this form.

Functions Defined Explicitly And Implicitly
 Explicit equations are those that are solved for y. We say “the
variable y is explicitly written function of x.
 Implicit equations are those not solved for y. As a result, we
must use or apply implicit differentiation.
y = 𝑥2 + 4𝑥 + 24
𝑥2 + 3𝑦3 + 4𝑦 = 2

 An equation of the form y = (𝑓 𝑥 ) is said to deﬁne y
explicitly as a function of 𝑥 because the variable 𝑦 appears
alone on one side of the equation and does not appear at all on
the other side.
 However, sometimes functions are deﬁned by equations in
which y is not alone on one side; for example,
𝑦𝑥 + 𝑦 + 1 = 𝑥

 The equation is not of the form y = (𝑓 𝑥 ), but it still deﬁnes
y as a function of 𝑥 since it can be rewritten as
𝑦 =
𝑥 − 1
𝑥 + 1

Steps Of Implicit Differentiation
 Here are the steps for solving implicit differentiation
i. Differentiate both sides of equation with respect to x
ii. Apply the rules of differentiation if necessary.
iii. Isolate all terms with dy/dx
iv. Factor out dy/dx
v. Divide on both sides of the equation to isolate dy/dx

Examples
i.
𝑑
𝑑𝑥
𝑥2 + 𝑦2 = 100
ii.
𝑑
𝑑𝑥
2𝑥2 − 3𝑦3 = 5
iii.
𝑑
𝑑𝑥
tan 𝑥𝑦 = 7
iv.
𝑑
𝑑𝑥
36 = 𝑥2 + 𝑦2
v.
𝑑
𝑑𝑥
5𝑥𝑦 − 𝑦3 = 8
vi. 4𝑥2 − 2𝑦2 = 9 ′′

Examples
vii.
𝑑
𝑑𝑥
𝑥3 + 𝑥2 𝑦 + 4𝑦2 = 6
viii.
𝑑
𝑑𝑥
4 cos 𝑥 sin 𝑦 = 1

Classwork
i.
𝑑
𝑑𝑥
𝑥3 + 𝑦3 = 9
ii.
𝑑
𝑑𝑥
𝑥 + 𝑥𝑦 + 2𝑥3
= 2
iii.
𝑑
𝑑𝑥
2𝑥3 − 3𝑦2 = 4
iv.
𝑑
𝑑𝑥
5𝑦2
+ sin 𝑦 = 𝑥2
v.
𝑑
𝑑𝑥
(cos 𝑥𝑦2 = 10)

Classwork
vi.
𝑑
𝑑𝑥
10𝑥2 𝑦 + 4𝑦4 = 2000
vii.
𝑑
𝑑𝑥
5𝑦3 − 3𝑥4 + tan 𝑥 = 5𝑥

Ch 3 the derivative

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