Chapter 4
Differentiation
4.1 Definition of Function Derivative
Differentiation is a method to compute the rate at which a dependent variable
changes with respect to the change in the independent variable. This rate of change is
called the derivative of with respect to .
Definition of Derivative Function
The derivative of the function ( ) with respect to the variable is the function
whose value at is
( + ℎ) − ( )
( ) = lim
→ ℎ
provided the limit exists
There are many ways to denote the derivative of a function = ( ) where the
independent variable is and the dependent variable is . Some common alternative
notations for the derivative are
( )= = = = ( )= ( )( ) = ( ).
Example 4.1 Show that = | | is differentiable on (−∞, 0) and (0, ∞) but has no
derivative at the origin = 0.
Solution
To the right of the origin,
(| |) = ( ) = 1.
Mathematical Analysis 34
,To the left,
(| |) = (− ) = −1.
There can be no derivative at the origin because the one-sided derivatives differ there:
(0 + ℎ) − (0) |ℎ| − |0|
Right − hand derivative of | | at zero = lim = lim
→ ℎ → ℎ
ℎ
= lim = 1.
→ ℎ
(0 + ℎ) − (0) |ℎ| − |0|
Left − hand derivative of | | at zero = lim = lim
→ ℎ → ℎ
−ℎ
= lim = −1.
→ ℎ
4.2 Differentiation of the power function
[ ]= .
Example 4.2
(1) ( )= ⟹ ( )=4 .
(2) ( )= ⟹ ( )=8 .
4.3 Basic Derivative Rules
4.3.1 Constant Rule
The derivative of a constant is zero; that is, for a constant :
( )=0
4.3.2 Constant Multiple Rule
. ( ) = . ( ) .
Mathematical Analysis 35
, Example 4.3 Evaluate
3( + 1)
Solution
3( + 1) = 3. ( + 1) = 3. 2 = 6 .
4.3.3 Multiple Rule
( ) = . ( ) .
Example 4.4 Find the derivative of = 3 sin 3 .
Solution
(3 sin 3 ) = 3 . (sin 3 ) = 3.3 3 =9 3 .
4.3.4 Sum/Difference Rules
( )± ( ) = ( ) ± ( )
Example 4.5 Find the derivative of
( )= + 3 − 2 tan .
Solution
( + 3 − 2 tan ) = ( )+ (3 ) − (2 tan )
= ( ) + 3. ( ) − 2. (tan ) = 2 + 3 − 2 .
Mathematical Analysis 36
Differentiation
4.1 Definition of Function Derivative
Differentiation is a method to compute the rate at which a dependent variable
changes with respect to the change in the independent variable. This rate of change is
called the derivative of with respect to .
Definition of Derivative Function
The derivative of the function ( ) with respect to the variable is the function
whose value at is
( + ℎ) − ( )
( ) = lim
→ ℎ
provided the limit exists
There are many ways to denote the derivative of a function = ( ) where the
independent variable is and the dependent variable is . Some common alternative
notations for the derivative are
( )= = = = ( )= ( )( ) = ( ).
Example 4.1 Show that = | | is differentiable on (−∞, 0) and (0, ∞) but has no
derivative at the origin = 0.
Solution
To the right of the origin,
(| |) = ( ) = 1.
Mathematical Analysis 34
,To the left,
(| |) = (− ) = −1.
There can be no derivative at the origin because the one-sided derivatives differ there:
(0 + ℎ) − (0) |ℎ| − |0|
Right − hand derivative of | | at zero = lim = lim
→ ℎ → ℎ
ℎ
= lim = 1.
→ ℎ
(0 + ℎ) − (0) |ℎ| − |0|
Left − hand derivative of | | at zero = lim = lim
→ ℎ → ℎ
−ℎ
= lim = −1.
→ ℎ
4.2 Differentiation of the power function
[ ]= .
Example 4.2
(1) ( )= ⟹ ( )=4 .
(2) ( )= ⟹ ( )=8 .
4.3 Basic Derivative Rules
4.3.1 Constant Rule
The derivative of a constant is zero; that is, for a constant :
( )=0
4.3.2 Constant Multiple Rule
. ( ) = . ( ) .
Mathematical Analysis 35
, Example 4.3 Evaluate
3( + 1)
Solution
3( + 1) = 3. ( + 1) = 3. 2 = 6 .
4.3.3 Multiple Rule
( ) = . ( ) .
Example 4.4 Find the derivative of = 3 sin 3 .
Solution
(3 sin 3 ) = 3 . (sin 3 ) = 3.3 3 =9 3 .
4.3.4 Sum/Difference Rules
( )± ( ) = ( ) ± ( )
Example 4.5 Find the derivative of
( )= + 3 − 2 tan .
Solution
( + 3 − 2 tan ) = ( )+ (3 ) − (2 tan )
= ( ) + 3. ( ) − 2. (tan ) = 2 + 3 − 2 .
Mathematical Analysis 36
