ordinary differentiation

8/9/2019 Ordinary Differentiation

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ORDINARY DIFFERENTIATION

DIFFERENTIAL COEFFICIENT(Definition):

The limit of incremental ratio i.e. lim as approaches to zero is called the differential coefficient

of y with respect to x and denoted by .

Differential coefficient of :

Let (1)

Let be the increment in and the corresponding increment in .

Then

Subtracting (1) from (2)

Diving the above result by ,we get


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Differential coefficient of

Let (1)

Let is the corresponding increment in to the increment in

(2)

Subtracting the equation (1) from (2)

Dividing the above result by ,


Let .(1)

Let increment in is and the corresponding increment in is ,

.(2)

Subtracting the equation (1) from (2)


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Dividing the above result by ,we get

But , thus


Let ..(1)

Let the increment in is and the corresponding increment in is , then

(2)

Subtracting the equation (1) from (2) , we get

Dividing the above result by , we get

So

But


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So

Since therefore

Hence


Let (1)

.(2)

Subtracting (1) from (2) ,

Dividing the equation (3) by , we get

So

But

therefore


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Let .(1)

Let the increment in is and the corresponding increment in is ,then

(2)

Subtracting the equation (1) from (2),we get

Dividing both the sides of the above expression by , we get

As then , therefore

since


Let (1)

If is the increment in and the corresponding increment in y is , then


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(2)

Subtracting the equation (1) from (2), we get

Dividing the above equation by , we get

Since as , therefore

Since

Thus

Similarly we can find the differential coefficient of .


Let (1)

If is the increment in y corresponding to the increment in ,then

..(2)



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Or

Or

Or

Or

Or

Dividing by

Since as ,therefore

Since

Thus

Or


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Similarly we can find the differential coefficient of .


Let

.(1)If the increment in is and the corresponding increment in is , then

(2)

Subtracting the equation (1) from (2) , we get

Or

Multiplying the above by

Or

Or

Since as ,

Since

Thus

Or

But from the equation (1) , therefore


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Differential coefficient of the product of two functions

Let and are two functions of and

(1)

If is the increment in and , are the corresponding increments in

respectively, then

(2)

Subtracting the equation(1) from (2), we get

Dividing the above equation by , we get

Since as therefore

Differential coefficient of the quotient of two functions

Let


Or

Or


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Or

Dividing the above equation by ,we get

Thus

Or

Differential coefficient of a function of a function(Chain Rule)

Let be a function of and is a function of

Let and be the corresponding increments in and y respectively.

Since as , therefore

Thus,

Logarithmic Differentiation

This method is applied to the functions in the form

Step.1: First we take log on both the sides of the given equation, so we get

Step2: Then we differentiate the equation obtained in the step.1


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13. 14.

15. 16.

17. Product Rule: 18. Quotient rule: .

19. Chain Rule: If y is function of u and u is a function of x, then .

20. Leibnitzs Rule: If u and v are two function of same variable, then derivative of their product is

given by .

ordinary differentiation

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