Methods of Differentiation

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Unit 2

• Methods of Differentiation

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2.1 Fundamental Formulas for Differentiation
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2.1 Fundamental Formulas for Differentiation
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2.2 Rules for Differentiation of Composite
Functions and Inverse Functions
The chain rule
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2.2 Rules for Differentiation of Composite
Functions and Inverse Functions
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2.2 Rules for Differentiation of Composite
Functions and Inverse Functions
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2.2 Rules for Differentiation of Composite
Functions and Inverse Functions
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2.2 Rules for Differentiation of Composite
Functions and Inverse Functions
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2.3 The Number e
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2.4 Differentiation of Logarithmic and
Exponential Functions
The symbol logax denotes the logarithm of the
number x to the base a.
Properties of logarithm

• loga1 0 and logaa 1
• loga(xy) logax logay

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2.4 Differentiation of Logarithmic and
Exponential Functions
Properties of logarithm
(4) logaxn nlogax, for any real number n.
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2.4 Differentiation of Logarithmic and
Exponential Functions
Logarithm to the base e is called natural
logarithm or Napierian logarithm. The natural
logarithm of x is denoted by ln x or loge x or
log x.
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2.4 Differentiation of Logarithmic and
Exponential Functions
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2.4 Differentiation of Logarithmic and
Exponential Functions
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2.4 Differentiation of Logarithmic and
Exponential Functions
Readers should not make the following mistakes
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2.4 Differentiation of Logarithmic and
Exponential Functions
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2.4 Differentiation of Logarithmic and
Exponential Functions
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2.4 Differentiation of Logarithmic and
Exponential Functions
Graphs of Logarithmic Functions
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2.4 Differentiation of Logarithmic and
Exponential Functions
Graphs of Exponential Functions
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P.46 Ex.2A
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2.5 Differentiation of Trigonometric Functions
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2.5 Differentiation of Trigonometric Functions
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2.5 Differentiation of Trigonometric Functions
Differentiation of trigonometric functions
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2.5 Differentiation of Trigonometric Functions
If u is a differentiable function of x,
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2.6 The Inverse Trigonometric Functions
The function y sin-1x is equivalent to x siny.
The other five functions y cos-1x, y tan-1x,
y cot-1x, y sec-1x and y csc-1x are defined
similarly. These six functions are called inverse
trigonometric functions or inverse circular
functions.
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2.6 The Inverse Trigonometric Functions
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2.6 The Inverse Trigonometric Functions
Graphs of six inverse trigonometric functions
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2.6 The Inverse Trigonometric Functions
Graphs of six inverse trigonometric functions
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2.6 The Inverse Trigonometric Functions
Relationship between inverse function of sine and
cosine
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2.6 The Inverse Trigonometric Functions
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P.51 Ex.2B
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2.7 Differentiation of Inverse Trigonometric
Functions
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2.7 Differentiation of Inverse Trigonometric
Functions
If u is a differentiable function of x,
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P.54 Ex.2C
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2.8 Equations of Tangents and Normals to a Curve
Let P and Q be two points on a curve and suppose
a straight line is drawn through them, then the
limiting position PT of this straight line as Q
moves along the curve and approaches P as a
limit, is called a tangent to the curve at the
point P.
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2.8 Equations of Tangents and Normals to a Curve
The straight line line perpendicular to the
tangent at the point of contact is called the
normal to the curve at that point.
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2.8 Equations of Tangents and Normals to a Curve
The angles between two curves at that point of
intersection are the angles between the two
tangents to these curves at that point.
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2.8 Equations of Tangents and Normals to a Curve
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P.56 Ex.2D
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2.9 Second Derivatives
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P.58 Ex.2E
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2.10 Indeterminate Forms and LHospitalsRule
The limit of a non-constant function f(x) as x
tends to x0 is said to be an indeterminate of the
form.
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2.10 Indeterminate Forms and LHospitalsRule
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2.10 Indeterminate Forms and LHospitalsRule
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2.10 Indeterminate Forms and LHospitalsRule
LHospitals rule
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P.63 Ex. 2F
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2.11 Differential and Small Increment
Let y f(x). The increment ?x of the independent
variable x is called the differential of x,
denoted by dx. The differential of y or the
differential of f(x), denoted by dy or df(x)
respectively, is defined by dy df(x) f (x)?x
f (x)dx, where f (x) is the derivative of
f(x).
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P.68 Ex. 2G
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P.68 Ex. 2H