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 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
Instantaneous speed is the speed at a particular instant (when the interval of time is infinitely small).
i.e., instantaneous speed
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 …
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