THE DERIVATIVE AND DIFFERENTIATION
OF ALGEBRAIC FUNCTIONS
OBJECTIVES:
• to define the derivative of a function
• to find the derivative of a function by
increment method (4-step rule)
• to identify the different rules of differentiation
and distinguish one from the other;
• prove the different rules of differentiation using
the increment method;
• find the derivative of an algebraic function using
the basic rules of differentiation; and
• extend these basic rules to other “complex”
algebraic functions.
Derivative of a Function
The process of finding the derivative of a
function is called differentiation and the
branch of calculus that deals with this
process is called differential calculus.
Differentiation is an important mathematical
tool in physics, mechanics, economics and
many other disciplines that involve change
and motion.
Consider a point Q( x2 , f ( x2 )) on the curve
y  f (x), that is distinct from P( x1 , f ( x1 )),and
compute the slope mPQ of the secant line
through P and Q.
mPQ
f ( x2 )  f ( x1 )

x
mPQ
where : x  x2  x1
and x2  x1  x
f ( x1  x)  f ( x1 )

x
If we let x2 approach x1, then the point Q
will move along the curve and approach
point P. As point Q approaches P, the value
of Δx approaches zero and the secant line
through P and Q approaches a limiting
position, then we will consider that position
to be the position of the tangent line at P.
y
tangent line
P( x1 , f ( x1 ))
Q( x2 , f ( x2 ))
secant line
y
y  f (x)
x
x  x 2  x1
x 2  x1  x
Thus, we make the following definition.
DEFINITION:
Suppose that x1 is in the domain of the function
f, the tangent line to the curve y=f(x) at the point
P(x1,f(x1)) is the line with equation,
y  f ( x1 )  m( x  x1 )
f ( x1  x )  f ( x1 )
where m  lim
provided
x 0
x
the limit exists, and P( x1 , f ( x1 )) is the point
of tangency.
DEFINITION
The derivative of y = f(x) at point P on the
curve is equal to the slope of the tangent line at
P, thus the derivative of the function f given by
y= f(x) with respect to x at any x in its domain is
defined as:
dy
y
f ( x  x)  f ( x )
 lim
 lim
dx x 0 x x0
x
provided the limit exists.
Other notations for the derivative of a function are:
d
D x y, D x f ( x), y ' , f ' , f ' ( x), and
f ( x)
dx
Note:
To find the slope of the tangent line to the curve at point
P means that we are to find the value of the derivative at
that point P.
There are two ways of finding the derivative of a function:
1. By using the increment method or the four-step rule
2. By using the differentiation formulas
THE INCREMENT METHOD OR THE FOUR-STEP RULE
One method of determining the derivative of
a function is the increment method or more
commonly known as the four-step rule.
The procedure is as follows
:
STEP 1: Substitute x + Δx for x and
y + Δy for y in y = f(x)
STEP 2: Subtract y = f(x) from the result of
step 1 to obtain Δy in terms of x
and Δx
STEP 3: Divide both sides of step 2 by Δx.
STEP 4: Find the limit of step 3 as Δx
approaches 0.
EXAMPLE
dy
1. Find
u sin g the four  step rule given y  1  x 2 .
dx
2
y
a. y  y  1   x  x 
d . lim
 lim  2 x  x 
x 0 x
x 0
2
b. y  y  y  1   x  x   y
dy
2
 2 x
2
y  1  x  2 xx  x  y
dx
y  1  x  2 xx  x  1  x 2 
2
2
2
y  1  x  2 xx  x  1  x 2
2
y  2 xx  x
2
y x 2 x  x 
c.

x
x
EXAMPLE 2. Find dy u sin g the four  step rule given y  2 x  1
dx
a. y  y 
1  2x
2( x  x )  1
1  2( x  x )
2( x  x )  1 2 x  1

1  2( x  x ) 1  2 x
2 x  2x  11  2 x   2 x  11  2 x  2x 
y 
1  2 x  2x 1  2 x )
b. y 
2 x  2x  1  4 x 2  4 xx  2 x  2 x  4 x 2  4 xx  1  2 x  2x
y 
1  2 x  2x 1  2 x )
c.
y
4 x

x x( 1  2 x  2x )( 1  2 x )
y
4
 lim
x 0 x
x 0 ( 1  2 x  2 x )( 1  2 x )
d . lim
dy
4


dx ( 1  2 x )2
EXAMPLE
dy
3. Find
u sin g the four  step rule given y  x  1 when x  10
dx
a . y  y  x  x  1
when x  10,
b. y 
dy
1
1
1



dx 2 10  1 2( 3 ) 6
x  x  1  x  1
y
x  x  1  x  1

x
x
y
x  x  1  x  1
x  x  1  x  1
d. lim
 lim

x 0 x
x 0
x
x  x  1  x  1
x  x  1  x  1
 lim
x 0 x
x  x  1  x  1
x
 lim
x 0 x
x  x  1  x  1
y
1
lim
 lim
x 0 x
x 0
x  x  1  x  1
dy
1
 
dx 2 x  1
c.



