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

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.

• 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)

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
• 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
• (Power Rule) For any rational number n,
• where u is a differentiable function of x and u(x)≠0.
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
• 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
• 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…
• Furthermore, it can be shown (in Chapter 7 and 8) that:
• (1)
• (2)
• Define y = ex and lnx = logex.
Differentiation of Logarithmic function f(x) = lnx

Proof:

Proof:

By Chain Rule and Formula 4.1

Differentiation of Logarithmic and Exponential Functions

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

Logarithmic Differentiation

Examples

Section 5Differentiation of

Trigonometric Function

Proof of Formula

Graphs

Proof of Formula

Section 10 Indeterminate Forms and L’Hospital Rule

Indeterminate Forms

(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

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) = 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