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Differentiation

Derivative of a Function Using First Principle

• Suppose f is a real-valued 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 real-valued 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) = cosec2 2x + tan2 4x. Also, find at x = .

Solution:

The derivative of f(x) = cosec2 2x + tan2 4x is calculated as follows.

At x = , is given by

Example 2:

If y = (ax2 + x + b)2, then find the values of a and b,such that .

Solution:

It is given that y = (ax2 + x + b)2

Comparing the coefficients of x3, x2, 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

Derivatives of Trigonometric Functions and Standard Formulas

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