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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 Δxy 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 + Δxy + 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: $\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}$

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 …

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