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 …
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