C2 methods of differentiation
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C2 Methods of Differentiation. Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit. provided this limit exists. Recall. Section 1. Fundamental Formulas for Differentiation. Formula 1.1 The derivative of a constant is 0.

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C2 Methods of Differentiation

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C2 methods of differentiation

C2 Methods of Differentiation


Recall

  • Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit

  • provided this limit exists.

Recall


Section 1 fundamental formulas for differentiation

Section 1. Fundamental Formulas for Differentiation

  • Formula 1.1

    The derivative of a constant is 0.

  • Formula 1.2

    The derivative of the identity function f(x)=x is the constant function f'(x)=1.

  • Formula 1.3

    If f and g are differentiable functions, then

    (f±g)'(x)= f'(x)±g'(x)


C2 methods of differentiation

  • Corollary 1.4

    (u1+u2+…+un)’= u1’+u2’+…+un’

  • Formula 1.5 (The product rule)

    (fg)'(x) = f(x) g'(x) + g(x) f'(x)

  • Corollary 1.6

    (u1×u2×…×un)’

    = u2×…×un×u1’+ u1u3×…×un×u2’+ u1u2u4×…×un×u3’

    +…+ u1×u2×u3×…×un-1×un’

  • Corollary 1.7

    (cu)’ = cu’

  • Formula 1.8


2 rules for differentiation of composite functions and inverse functions

2. Rules for Differentiation of Composite Functions and Inverse Functions

  • Formula 2.1 (The Chain Rule)

    Let F be the composition of two differentiable functions f and g;

    F(x) = f(g(x)).

    Then F is differentiable and

    F'(x) = f'(g(x)) g'(x)

    Proof:

    Exercise


Formula 2 2

Formula 2.2

  • (Power Rule) For any rational number n,

  • where u is a differentiable function of x and u(x)≠0.


C2 methods of differentiation

  • Corollary 2.3 For any rational number n,

    if f(x)=xn where n is a positive integer, then

    f'(x)= nxn-1


Formula 2 4

Formula 2.4

  • If y is differentiable function of x given by y=f(x), and if x=f –1(y) with f’(x) ≠0, then

  • Practice


Section 3 the number e

Section 3 The Number e

  • A man has borrow a amount of $P from a loan shark for a year. The annual interest rate is 100%. Find the total amount after one year if the loan is compounded :

  • (a) yearly;(b) half-yearly

  • (c) quarterly(d) monthly;

  • (e) daily;(f) hourly;

  • (g) minutely;(h) secondly.

  • (h) Rank them in ascending order.

  • (i) Will the amount increase indefinitely?AnswersGraphs


E 2 718281828459045

e= = 2.718281828459045…

  • Furthermore, it can be shown (in Chapter 7 and 8) that:

  • (1)

  • (2)


Section 4 differentiation of logarithmic and exponential functions

Section 4 Differentiation of Logarithmic and Exponential Functions

  • Define y = ex and lnx = logex.


Differentiation of logarithmic function f x lnx

Differentiation of Logarithmic function f(x) = lnx

Proof:

Proof:

By Chain Rule and Formula 4.1


Differentiation of logarithmic and exponential functions

Differentiation of Logarithmic and Exponential Functions

  • Exercises on

  • Product Rule

  • Quotient Rule

  • Chain Rule


Logarithmic differentiation

Logarithmic Differentiation

Examples

Read Examples 4.2- 4.4


Formula 4 4

Formula 4.4


Formula 4 5

Formula 4.5

Quiz


C2 methods of differentiation

Section 5Differentiation of

Trigonometric Function

Proof of Formula


Graphs of trigonometric functions

Graphs of trigonometric functions


Section 6 the inverse trigonometric functions

Section 6 The Inverse Trigonometric Functions


Y cosx and y arccosx

y=cosx and y=arccosx


Y tanx and y arctanx

y=tanx and y=arctanx


Y cotx and y arccotx y secx and y arcsecx y cscx and y arccscx

y=cotx and y=arccotxy=secx and y=arcsecxy=cscx and y=arccscx

Graphs


Section 7 differentiation of inverse of trigonometric function

Section 7Differentiation of Inverse of Trigonometric Function

Proof of Formula


Section 10 indeterminate forms and l hospital rule

Section 10 Indeterminate Forms and L’Hospital Rule

Indeterminate Forms


C2 methods of differentiation

(i) Evaluate limx→a f(x)/g(x) where f(a)=g(a)=0.

1.Evaluate limx→o sin3x/sin2x.

L’Hospital:

limx→o sin3x/sin2x

= limx→o 3cos3x/2cos2x

= 3/2

2.limx→o (x-sinx)/x3=limx→o (1-cosx)/3x2

= limx→o(sinx)/6x

= limx→o(cosx)/6

= 1/6

How?

Why?


Proof of 0 0

Proof of 0/0

limx→af(x)/g(x)

= limx→a(f(x) – f(a))/(g(x) – g(a))

= limx→a(f(x) – f(a))/(x-a)/(g(x) – g(a))/(x-a)

= (limx→a(f(x) – f(a))/(x-a))/( limx→a (g(x) – g(a))/(x-a))

= f’(a)/g’(a)


Differentiation of exponential function f x e x

Differentiation of exponential function f(x) = ex

  • Theorem. Let f(x)=bx be the exponential function. Then the derivative of f is

    f'(x) = bx f'(0)

  • Proof

  • Hope: e is the real number such that the slope of the tangent line to the graph of the exponential function y=ex at x=0 is 1.

  • Formula 4.3 Let f(x)=ex be the exponential function. Then the derivative of f is

    f'(x) = ex


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