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 $\mathrm{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 positiontime graph, the slope of the curve indicates the velocity and the angle of the slope with the xaxis indicates the direction.

Instantaneous velocity is the velocity at a particular instant (slope at a particular point on the xt curve).
Derivative/ Differentiation from the first principal

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.
 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.
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
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 poinâ€¦
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