Pascal's Triangle & Binomial Theorem

Welcome To My Lecture

Differentiation by Protima Dash

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Differentiation

• The derivative is a mathematical operator, which measures the rate of change of a quantity relative to another quantity. The process of

finding a derivative is called differentiation.

• There are many phenomena related changing quantities such as speed of a particle, inflation of currency, intensity of an earthquake and voltage of an electrical signal etc. in the world. In this chapter we will discuss about various techniques of derivative

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Example:

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�

�� ( 1 ) = 0 ;

Constant Rule:

If is a constant, then

�

�� ( � ) = 0 ;

�

�� ( 1000 ) =0 ;

�

�� ( � ) =0.

Example: Find for the function

 

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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

 

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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

 

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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

 

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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

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Quotient Rule:

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

Example: Find for the function

 

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

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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

��

�� = ��

�� ∙ ��

��

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Find for the functions and

 

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,

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Function as Power of another Function:

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

Example: Find for the function

 

Given that,

�

��

(

� ^�

)

+ �

��

(

� ^�

)

= �

�� (1)

Taking derivative with respect to on both sides,

⇒ �^� �

�� ( � ln �)+ �^� �

�� (� ln �)=0

⇒�

^�

( ^{� 1} � +ln � ��

�� ) ^{+ �}

^�

( ^� � ¹

��

�� + ln � ) ⁼⁰

⇒�

^�

� �

⁻¹

+ �

^�

ln � ��

�� + �

^�

� �

⁻¹

��

�� + �

^�

ln � = 0

⇒ �^�−1 �+�^� ln � ��

�� + �^�−1� ��

�� + �^� ln �=0

�

�� ( �� ) =� �

�� ( � ) + � �

�� ( � )

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Function as Power of another Function:

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

��

��

(

�^� ln �+ �^�−1�

)

=−�^�−1� − �^� ln �

∴ ��

�� =−

(

_�^�−1 _� +�^��� �

)

�^� �� �+ �^�−1 �

��

�� = −�^�−1 � −�^� ln �

�^� ln �+ �^�−1�

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(a) Find the derivative of the following functions :

(b) Find of the parametric equations:

and ^.

Exercise