In somewhat dated language, differentiation expresses the rate at which one quantity y changes as a result of a change in another quantity x on which it has a functional relationship. Using the symbol Δ to refer to change in a quantity, this rate is defined as a limit of difference quotients

as Δx approaches 0. In Leibnitz' notation, the derivative of y with respect to x is written

suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as dy over dx. The form dy dx is also used conversationally, although it may be confused with the notation for element of area.

In contemporary mathematical language, one dispenses with referring to dependent quantities and simply states that differentiation is a mathematical operation on functions. The precise definition of this operation (which also dispenses with referring to infinitesimal quantities) is given as the limit as h approaches 0 of

This definition is discussed in more detail below. If f is a function, the derivative of the function f at the value x is written in several ways:

pronounced f prime of x

pronounced d by d x of f of x or d d x of f of x.

pronounced d f by d x or d f d x

A function is differentiable at a point x if its derivative exists at that point; a function is differentiable on an interval if it is differentiable at every x within the interval. If a function is not continuous at c, then there is no slope and the function is therefore not differentiable at c; however, even if a function is continuous at c, it may not be differentiable.

The derivative of a differentiable function can itself be differentiable. The derivative of a derivative is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on.




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