Differentiation
Derivative of a Function Using First Principle

Suppose f is a realvalued function and a is a point in the domain of definition. If the limit exists, then it is called the derivative of f at a. The derivative of f at a is denoted by.
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 Suppose f is a realvalued function. The derivative of f {denoted by or } is defined by
This definition of derivative is called the first principle of derivative.
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For example, the derivative of is calculated as follows.
We have; using the first principle of derivative, we obtain
Solved Examples
Example 1:
Find the derivative of f(x) = cosec^{2} 2x + tan^{2} 4x. Also, find at x = .
Solution:
The derivative of f(x) = cosec^{2} 2x + tan^{2} 4x is calculated as follows.
At x = , is given by
Example 2:
If y = (ax^{2} + x + b)^{2}, then find the values of a and b,such that .
Solution:
It is given that y = (ax^{2} + x + b)^{2}
Comparing the coefficients of x^{3}, x^{2}, x, and the constant terms of the above expression, we obtain
Example 3:
What is the derivative of y with respect to x, if?
Solution:
It is given that

Suppose f is a realvalued function and a is a point in the domain of definâ€¦
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