Definition: Derivative

Let be a real valued continunes function of . Then a change (increment) in the value of variable will bring about a change in the invariable . These changes are denoted by respectively. (delta and delta )

Then

Hence the change in . The average rate of change of for a change in is If the limit of this ratio as exists, then we call that limit as the differential coefficient or the derivative of with respect to and is denoted by or .

Other notations are etc.

Thus ,the process of finding the derivative is called differentiation.

We thus see that derivative is a measure of rate of change of one variable with respect to change in another. The rate of change of with respect to is

or

[Note: In the second formula, the increment in in is ].

An important point to remember is that the symbol is not to be meant as the quotient of dy and dx but the result of the operation of differentiation of y with respect to x. The symbol denotes the differential operator.

Geometrical meaning of derivative

Let be any fixed point on a curve and be any other point which is close to the point .

Let be and be . Join . Draw and perpendicular to the -axis and .
It is evident that and
In
Let
Then the slope(gradient) of (say ) is given by
As also tends to zero and the secant of the curve becomes the tangent at to the curve.
(refer to definition of derivative)
represents the slope of the tangent to the curve at the point .

Differentiation Techniques

1. Using definition of derivative, we can find the derivative or differential coefficient of a function This is known as finding derivative from the first principles. To carry out this the following steps are to be followed.
1. Increase the argerment by .
2. Calculate the increased value of the function
3. Calculate
4. Find the ratio
5. Allow and find the .
This gives the general form or of the given function .
1. To find the derivative of a function at a point  in its domain, use the formula
[Here is the increment given to the value of
Example: 1
Find the value of derivative at  of the function

Solution:

Example: 2
Find the derivative of a constant function.

Solution:
Let (a constant)

Example: 3
Differentiate from first principles.

Solution:

Example: 4
Find the derivative from the first principles.

Solution: