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Derivatives

Derivative of a Function Using First Principle

Derivative as a Rate Measurer


Let x and y be two quantities interrelated in such a way that for each value of x there is one and only one value of y.

The graph represents the y versus x curve. Any point in the graph gives unique values of x and y. Let us consider point A on the graph. We shall increase x by a small amount Δx, and the corresponding change in y be Δy.

Thus, when x change by Δxy change by Δy and the rate of change of y with respect to x is equal to 

In the triangle ABC, the coordinates of A is (x, y); coordinate of B is (x + Δxy + Δy)

The rate  can be written as,

But this cannot be the precise definition of the rate because the rate also varies between the point A and B. So, we must take a very small change in x. That is Δx is nearly equal to zero. As we make Δx smaller and smaller the slope tanθ of the line AB approaches the slope of the tangent at A. This slope of the tangent at A gives the rate of change of y with respect to x at A.

This rate is denoted by 

and, 



Note: dydx=1dxdy


Speed

  •  Speed = 

  • Instantaneous speed is the speed at a particular instant (when the interval of time is infinitely small).

          i.e., instantaneous speed 

Velocity

  • Velocity = 

  • In a position-time graph, the slope of the curve indicates the velocity and the angle of the slope with the x-axis indicates the direction.


 
  • Instantaneous velocity is the velocity at a particular instant (slope at a particular point on the x-t curve).

         


Derivative/ Differentiation from the first principal

  • 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.

  • 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.

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

Derivative as a Rate Measurer


Let x and y be two quantities interrelated in such a way that for each value of x there is one and only one value of y.

The graph represents the y versus x curve. Any point in the graph gives unique values of x and y. Let us consider poin…

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