- 60 Views
- Uploaded on
- Presentation posted in: General

C2 Methods of Differentiation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

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.

- 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

- 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

- (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

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

- 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

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

- Define y = ex and lnx = logex.

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

Read Examples 4.2- 4.4

Quiz

Section 5Differentiation of

Trigonometric Function

Proof of Formula

Graphs of trigonometric functions

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

Graphs

Section 7Differentiation of Inverse of Trigonometric Function

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?

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)

- 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