C2 Methods of Differentiation

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C2 Methods of Differentiation
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Recall

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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)

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• Corollary 1.4
• (u1u2un) u1u2un
• Formula 1.5 (The product rule)
• (fg) '(x) f(x) g '(x) g(x) f '(x)
• Corollary 1.6
• (u1u2un)
• u2unu1 u1u3unu2 u1u2u4unu3
• u1u2u3un-1un
• Corollary 1.7
• (cu) cu
• Formula 1.8

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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

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Formula 2.2

• (Power Rule) For any rational number n,
• where u is a differentiable function of x and
  u(x)?0.

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• Corollary 2.3 For any rational number n,
• if f(x)  xn where n is a positive integer, then
  
• f '(x)  n xn - 1

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Formula 2.4

• If y is differentiable function of x given by
  yf(x), and if xf 1(y) with f(x) ?0, then
• Practice

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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? Answers Graphs

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e 2.718281828459045

• Furthermore, it can be shown (in Chapter 7 and 8)
  that
• (1)
• (2)

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Section 4 Differentiation of Logarithmic and
Exponential Functions

• Define y ex and lnx logex.

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Differentiation of Logarithmic function f(x) lnx

• Proof

Proof By Chain Rule and Formula 4.1
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Differentiation of Logarithmic and Exponential
Functions

• Exercises on
• Product Rule
• Quotient Rule
• Chain Rule

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Logarithmic Differentiation

• Examples

Read Examples 4.2- 4.4
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Formula 4.4
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Formula 4.5
Quiz
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Section 5Differentiation of Trigonometric
Function
Proof of Formula
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Graphs of trigonometric functions
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Section 6 The Inverse Trigonometric Functions
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ycosx and yarccosx
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ytanx and yarctanx
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ycotx and yarccotx ysecx and
yarcsecx ycscx and yarccscx

• Graphs

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Section 7Differentiation of Inverse of
Trigonometric Function

• Proof of Formula

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Section 10 Indeterminate Forms and LHospital Rule

• Indeterminate Forms

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

• 1. Evaluate limx?o sin3x/sin2x.
• LHospital
• 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?
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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)

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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