Derivatives and Differentiability

In calculus (a branch of mathematics) the derivative is a measure of how a function changes as its input changesA function "f" in interval [a,b] is said to be a continuous function when the Graph drawn for f(x) is a smooth line or curve without any break in it.

Such curve or line can be drawn by the continuous motion of a pencil in a sheet of paper.

And Discontinuous function is just opposite of the continuous function , the function "f" is said to be discontinuous function when the graph drawn for f(x) is consists of disconnected curves or lines.

Conversely, the integral of the velocity over time is how much the vehicle's position changes from the time when the integral begins to the time when the integral ends.

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function. A function f is said to be differentiable at a if the limit of the difference quotient exists.

The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration.

A function "f" in interval [a,b] is said to be a continuous function when the Graph drawn for f(x) is a smooth line or curve without any break in it.

Such curve or line can be drawn by the continuous motion of a pencil in a sheet of paper.