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Differentiation (Pembezaan) Introduction

# Differentiation (Pembezaan) Introduction

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11/07/2012

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DIFFERENTIATION(PEMBEZAAN)INTRODUCTION
Calculus is divided into two main parts: differentiationand integration. Differentiation is sometimes referredto as differentiation calculus.We will apply some differentiation rules to simplify theprocess of finding the derivative of a function.
BASIC RULES OF DIFFERENTIATION
There are many rules of differentiation. However inthis chapter, we only cover the basic rules odifferentiation and apply it to the field of our study. The first rule of differentiation merely says that thatgiven any constants, its derivative is zero.
Example 1
a.
If y = 5, then dy = 0dx
b.
If y = √104, then d (√104) = 0dx
c.
If y = 1000
999
, then dy = 0Dx
Exercise 1
Find f’(x) if
a.f(x) = 8
Solution:
b.f(x) = -49
Solution:

RULE 1: DERIVATIVE OF CONSTANT
If f(x) = k, then d (k) = 0dx

c.f(x) = 2
Solution:
a.
f(x) = 20
48
Solution:
Example 2:
a.
If y =2x,dy = 2dxb.If f(x) = -12xf’(x) = -12
Exercise 2
Find f(x) if (a) f(x) = 7x(b) f(x) = -25xNext we find the derivative of a function of the form xraised to some constant power, written as x
n
. Suchfunction is called a power function. The derivative of the function is called the power rule.

RULE 2: CONSTANT – MULTIPLE RULE
If f(x) = mx, then f’(x) = m
RULE 3: POWER RULE
If f(x) = x
n
, then d (x
n
) = n.x
n-1
dx

In other words, the derivative of a constant power of xis equal to the exponent (power) times x to a powerone less than the old exponent.
Example 3:
Differentiation the following functions:(note: to differentiation a function also means to findthe derivative of the function)Old exponent times xOld exponent minus one
a.
If y =x
2
,dy = 2x
2-1
dx= 2x
b.
If y =
3x
y = x
1/3
dy = 1 x
1/3 -1
dx 3= 1 x
-2/3
3Exercise 3:Find f’(x) if (a) f(x) = ax
n
(b) f(x) = 9x
4

RULE 4 : CONSTANT FACTOR RULE
If y = kx
n
, where k is constant, then d kf(x)= n.ax
n-1
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