Pascal's Triangle & Binomial Theorem

Senior Lecturer Department of GED

Daffodil International University

Differentiation (Part – 3)

(Technique of Differentiation)

Example:

2

�

�� ( 1 ) = 0 ;

 

Constant Rule:

If is a constant, then

 

�

�� ( � ) = 0 ;

 

�

�� ( 1000 ) =0 ;

 

�

�� ( � ) =0.

 

Example: Find for the function

 

3

Given that,

 

��

�� = �

�� ( √ � )

 

Taking derivative with respect to on both sides,

 

¿ �

��

(

�

1 2

)

 

¿ 1 2 �

1 2 −1

 

��

�� = 1 2 �⁻

1

  2

∴ ��

�� = �

� √ �

 

¿ 1 2

1

�

1 2

 

Power Rule:

 

Sum or Difference Rule:

If and are functions of x, ( and ), then

 

4

Given that,

 

�

�� ( �

^�

) = � �

^�−1

 

��

�� = �

��

( √3 � − √1� )

 

Taking derivative with respect to on both sides,

 

¿ �

��

(

�

1

3

)

− �

��

(

�⁻

1 2

)

 

¿ 1 3 �

1 3−1

−

(

^{− 1}2 �⁻

1 2−1

 

)

��

�� =1

3 �⁻

2

3+ 1

2 �⁻

3

  2

∴ ��

�� = 1 3

1

√

3

�

²

+

1 2

1

√ �

³

 

¿ 1 3

1

�

2 3

+ 1 2

1

�

3 2

 

Example 1: Find for the function

 

Example 2: Find for the function  

5

Given that,

 

�

�� ( ��� � ) = 1

�

 

��

�� = �

��

(

^{��+2��� �− 1}2 ��� �

)

 

Taking derivative with respect to on both sides,

 

¿ �

��

(

�^�

)

+ �

�� (2 ��� �)− �

��

(

¹2 ��� �

)

 

¿ �

^�

+ 2 �

�� ( ��� � ) − 1 2

�

�� ( ��� � )

 

∴ ��

�� = �

^�

+ 2 cos � − 1 2 �

 

¿ �

^�

+ 2 cos � − 1 2

1

�

 

�

�� ( �

^�

) = �

^�

 

�

�� ( sin � ) = cos �

 

�

�� ( � ± � ) = �

�� ( � ) ± �

�� ( � )

 

Sum or Difference Rule: (Cont...)

�

�� ( �� ) = � �

�� ( � )

 

Example: Find for the function

 

6

Given that,

 

��

�� = �

�� ( �

²

���

⁻¹

� )

 

Taking derivative with respect to on both sides,

 

¿ �

²

�

�� ( ���

⁻¹

� ) + ���

⁻¹

� �

�� ( �

²

)

 

¿ �² 1

1+�² +���⁻¹ � . (2� )

 

∴ ��

�� = �^�

�+ �^� +�� tan^−��

 

�

��

(

���⁻¹ �

)

^{= 1}

1+ �²

 

 

Product Rule:

If and are functions of x, ( and ), then

 

Example: Find for the function

 

7

Quotient Rule:

If and are functions of x, ( and ), then

 

Given that,

 

��

�� = �

��

(

5−³��� �^�

)

 

Taking derivative with respect to on both sides,

 

¿

(5−��� �) ^�

�� (3 �)−3 � �

�� (5−��� �) (5−��� �)²

 

¿

(5−��� �)3 �

�� ( �)−3 � �

�� (5−��� �) (5−��� �)²

 

∴ ��

�� = ��−���� �+� � ���^��

(

�−��� �

)

^�

 

�

�� ( � )=1

 

�

�� ( ��� � ) = ���

²

�

 

��

�� = (_{5−��� �})₃(₁)_{− 3�}

(

0− ���²�

)

(_{5−��� �})²

 

�

�� ( � ) =0

 

8

Chain Rule:

If is a function of , () and is a function of x, (), then

 

Example: Find for the function

 

Given that,  

��

�� = �

�� (�^�)∙ �

��

(

2���⁻¹�

)

 

By Chain Rule,

¿�^�∙2 �

��

(

���⁻¹�

)

 

∴ ��

�� =−��^{����−��}

√

^{�− ��}

 

�

�� ( �

^�

) = �

^�

 

�

�� ( cos

⁻¹

� ) = − 1

√ 1 − �

²

 

��

�� =

(

_�^{2���−1 �}

)

∙2 −1

√

^{1− �2}

 

Suppose

 

So

 

��

�� = ��

�� ∙ ��

��

 

Find for the functions and

 

9

Given parametric equations, and

 

��

� � = �

� � ( � cos � +� sin � )

 

¿ �

� � ( � cos � ) + �

� � ( � sin � )

 

¿ � �

�� ( cos � ) + � �

� � ( sin � )

 

��

�� = ��

� � ∙ � �

�� =� cos � ∙ 1

� cos � − asin �

 

� �

� � = �

� � ( � sin � )

 

¿

 

a (− sin � )+ � cos �

∴ ��

� � = � cos � − asin �

 

Again , � =� sin �

 

¿ � �

� � ( sin � )

 

∴ � �

�� =� cos �

 

∴ ��

�� = � ��� �

���� � − ���� �

 

By Chain Rule,

Example: Find for the function

 

10

Function as Power of another Function:

If and are functions of x, ( and ), then

 

Given that,

 

�

��

(

�^�

)

+ �

��

(

�^�

)

= �

�� (1)

 Taking derivative with respect to on both sides,  

⇒ �^� �

�� (� ln �)+ �^� �

�� ( � ln �)=0

 

⇒ �^�

(

^{� 1}� +ln � ��

��

)

^{+ ��}

(

^� �¹

��

�� +ln �

)

⁼⁰

 

⇒ �^� � �⁻¹+�^� ln � ��

�� + �^� � �⁻¹ ��

�� + �^�ln �=0

 

��

��

(

�^� ln �+ �^�−1 �

)

=− �^�−1� − �^�ln �

 

∴ ��

�� =−

(

_�^�−1_� +� ^��� �

)

�^� �� �+ �^�−1 �

 

��

�� =−�^�−1 � −�^� ln �

�^� ln �+ �^�−1 �

 

⇒ �^�−1 �+�^� ln � ��

�� + �^�−1� ��

�� + �^� ln �=0

 

�

�� ( �� ) =� �

�� ( � ) + � �

�� ( � )

 

Exercise

11

(a) Find the derivative of the following functions :

(b) Find of the parametric equations:

and ^.