I have forgotten

• https://me.yahoo.com

Differentiation

Differentiation of Simple Algebraic Functions from first principles
View other versions (2)

Differentiation From First Principles

It is sometimes required that Differentiation be carried out from first principles.

Consider the following equation


Let there be small increase in x of  and let the corresponding increase in y be 

Rewriting the original equation


Multiplying out


But we know that


Subtracting equation (1) from Equation (2)


As the value  is reduced and tends towards dx i.e. until it is infinitesimally small, the value of  tends towards zero and can be neglected.  is now written as .

It can be seen from the diagram that the value of the tangent at x,y is  and at the limit this is written as 

Referring to Equation (1) the value of the tangent is given by:


 is known as the Gradient function and represents the Derivative of y with respect of x. It is also known as the b{Differential Coefficient}.

In the simplest case, if

then


Example:
Example - Differentiation
Problem
Find the differentiation of

Workings
Bringing the power of each x variable down, and subtracting 1 from each power of x yields:


which is simplified further to

Solution


The Differentiation 0f A Product Of Two Functions Of X

It is obvious, that by taking two simple factors such as 5 X 8 that the total increase in the product is Not obtained by multiplying together the increases of the separate factors and therefore the Differential Coefficient is not equal to the product of the d.c's of its factors. If

then


To Prove the Product Rule let  where u and v are both functions of x. Thus when x increases to  u and v will also change to  and .

Their product y will therefore become


Therefore , the increase in  Thus


In the limit as ,  and  tend to zero, so the above equation becomes:

Example:
Example - Differentiation - Product Rule
Problem
Differentiate

using the Product Rule
Workings
if

then the differential is


So for



Thus

Solution


The Differentiation Of A Product Of Any Number Of Functions Of X

The rule for finding the differential coefficient of a product of two functions of x can be extended to apply to the product of any finite numbers of functions of x If

Where u, v, w are all functions of x, then regarding this as the product of the two factors u and w:





And similarly for any finite number of factors.

Note An important result follows from the above rules. The differential coefficient of  with respect to x can be considered to be the product of two factors each of x and hence is given by:


Similarly, if n is any interger, by taking the product of n factors each of y

The differential coefficient of  With respect to 

The Differentiation Of A Quotient Of Two Functions Of X

Let
Then

The proof from first principles of the Quotient Rule.

As with previous proofs from first principles x becomes , u becomes  and v becomes  Therefore y becomes  Thus

Therefore


In the limit when  tends to zero the so will  and  Then

Example:
Example - Simple example
Problem
Differentiate using the quotient rule

Workings
Then

Solution