Differentiation of a Function

Derivative or differential coefficient of the function is an important branch of mathematics. Let y = f(x) be the real valued function. It means the value of y depends on the value of x and it changes with a change in the value of x.  So, x is called the independent variable and y is the dependent variable.  In simple terms, derivative of a function denotes how a quantity is changing with respect another quantity. 

Let ∆ x (or h) be the small increment in x and the corresponding change in y or f(x) be

∆y.  Then the value of  x  changes  from x  to x + ∆ x ( or h ) and the value of f(x) or y  changes from   f( x )  to f ( x + ∆ x).

So, change in the value of f(x) or ∆ y = f (x + ∆ x) – f(x).

Definition of derivatives or differentiation or differential coefficient of a function f (x):

 

Differentiation from first principles or ab – initio method or delta method:

The process of finding the derivative of a function   by using the definition of derivatives is called the differentiation from first principles. Using the first principle of differentiation, mathematicians have derived some standard formulas based on them; it becomes very easy to calculate differentiation of most of the functions.

Standard formulas

 

 

By using the above standard formulas, derivative of a function can be calculated as given below:

Some of the examples of solved problems are as below:

 Example 1:            x^y  =y^x

Solution:         Taking log on both the sides

           y log x = x log y

Differentiating both sides with respect to x:

Above is the final answer of the derivative of y with respect to x.

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