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 Δx, y 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 + Δx, y + 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 $\tan \theta$ 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: $\frac{dy}{dx}=\frac{1}{{\displaystyle \frac{dx}{dy}}}$

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…