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 …