DERIVATIVES

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CHAPTER 2

• DERIVATIVES

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CONTENT

• 2.1 Slope and equation of tangent
• 2.2 The derivative of a function
• 2.3 Techniques of Differentiation
• 2.4 Derivatives of Composite function (Chain
  Rule)
• 2.5 Derivatives of Trigonometric Function
• 2.6 Derivatives of Logarithmic Function
• 2.7 Derivatives of an Exponential Function
• 2.8 Implicit Function
• 2.9 Higher Derivatives

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OBJECTIVES

• By the end of this chapter student should able
  to
• Calculate derivative using the basic technique of
  differentiation
• Calculate derivative using Chain rule technique
• Calculate derivative for trigonometric,
  Logarithmic and Exponential function
• Calculate derivative for implicit functions
• Solve the higher derivative functions

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2.0 INTRODUCTION

• Calculus is the mathematics of change, and the
  primary tool for studying change is a procedure
  called differentiation (derivatives).
• Derivative
• derived a function.
• study on how one quantity changes in relation to
  another quantity.
• The derivatives of function f (x) study on how f
  (x) changes in relation to changes in x.

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2.1 SLOPE AND EQUATION OF TANGENT

• Consider a curve y f (x). Let P(x0,y0) be fixed
  point on the curve and let Q(x1,y1) be a nearby
  movable point on that curve.
• Slope of PQ (secant line)
• If Thus,

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Definition

• Example 2.1.1
• Find the slope of the parabola at
  the point (2, 5). Then find an equation for the
  tangent to the parabola at this point.

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Exercise 2.1

• Find an equation of the tangent line to the
  parabola at point (3, -6).
• Find an equation of the tangent line to the
  hyperbola at point (3, 1).
• Find the slopes of tangent lines to the graph of
  the function at point (1, 1), (4,
  2), and (9, 3).

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2.2 DERIVATIVES OF A FUNCTION

• The derivative of the function f (x) with respect
  to the variable x is the function whose value at
  x is
• Provided the limit exists. If exists, we
  say that f is differentiable at x.
• Generally, equation () is also called as
  differentiation with first principle.

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Notation
derivative of y with respect to x
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Finding Derivatives from the Definition ( First
Principle)

• Write expression for f (x) and f (x h)
• Find and simplify the difference quotient
• Evaluate
• If the limit does exist, then

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Example 2.2.1

• Use the definition to find the derivative of
  function,
• f (x) 3x

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Example 2.2.2

• Use the definition to find the derivative of
  function,

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Example 2.2.3

• Use the definition to find the derivative of
  function,

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Example 2.2.4

• Use the definition to find the derivative of
  function,

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Example 2.2.5

• Use the definition to find the derivative of
  function,

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Exercise 2.2

• Find the derivative of the following function by
  using first principle definition.

A B C
D E F
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2.3 TECHNIQUES OF DIFFERENTIATION
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Constant Function
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Identity Function
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Power Rule
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Power Rule
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Constant Multiple
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Sum Rule
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Difference Rule
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Linearity Rule
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Product Rule
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Quotient Rule
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Exercise 2.3

• Find the derivative of the following function

A B C
D E F
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Exercise 2.3

• Find the derivative of the following function

G H I
J K L
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2.4 DERIVATIVES OF COMPOSITE FUNCTION (CHAIN RULE)
If y f (u) and u g (x), the composite
function y is given by
Definition if f differentiable at x and g
differentiable at u g (x), then the composite
function
differentiable at x where
or
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Example 2.4.1 Find the derivative of the
following functions
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Example 2.4.1 Find the derivative of the
following functions
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Example 2.4.2 Chain Rule
If y f (u), u g (v), and v h (x) then
Find the
if
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Exercise 2.4

• Find the derivative of the following function by
  using chain rule

A B C
D E F
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2.5 DERIVATIVES OF TRIGONOMETRIC FUNCTION
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2.5 DERIVATIVES OF TRIGONOMETRIC FUNCTION
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Example 2.51 Find the derivative of y with
respect to x for the following function
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Example 2.51 Find the derivative of y with
respect to x for the following function
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Example 2.52 Find the derivative of y with
respect to x for the following function
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Example 2.52 Find the derivative of y with
respect to x for the following function
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Exercise 2.5

• Find the derivative of the following function

A B C D
E F G H
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2.6 DERIVATIVES OF LOGARITHMIC FUNCTION
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2.6 DERIVATIVES OF LOGARITHMIC FUNCTION
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Example 2.6 Differentiate the following function
with respect to x
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Example 2.6 Differentiate the following function
with respect to x
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Exercise 2.6

• Find the derivative of the following function

A B C D
E F G H
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2.7 DERIVATIVES OF AN EXPONENTIAL
FUNCTION
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Example 2.7 Differentiate the following function
with respect to x
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Exercise 2.7

• Find the derivative of the following function

A B C D
E F G H
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2.8 IMPLICIT FUNCTION

• Implicit function is a function of y that cannot
  be written directly with respect to x.
• An implicit function F is usually defined as
  
• Example of implicit function is as follows
• The implicit function can be differentiate one
  variable at a time with an assumption y is the
  function of x.

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Example 2.7 Differentiate the following function
with respect to x by using implicit
differentiation
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Exercise 2.8

• Find the derivative of the following function by
  using implicit differentiation

A B C D
E F G H
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2.9 HIGHER DERIVATIVES
first derivatives
Higher derivatives
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Example 2.9.1 2.9.2

• Given Find
• Find if

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Example 2.9.3

• By differentiating
• implicitly.
• Find

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Exercise 2.9

• Find the 1st derivative and 2nd derivatives of
  the following function

A B C
D E F
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SUMMARY
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SUMMARY
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SUMMARY
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SUMMARY
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SUMMARY
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Thank You