Derivative

In mathematics, the derivative is one of the two central concepts of calculus. (The other is the integral; the two are related via the fundamental theorem of calculus.)

Differentiation and differentiability

In physical terms, 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

Related Topics:
Functional relationship - Limit

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: rac{Delta y}{Delta x}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

as Δx approaches 0. In Leibniz's notation for derivatives, the derivative of y with respect to x is written

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: rac{dy}{dx}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. The precise definition of this operation (which therefore need not deal with infinitesimal quantities) is given as:

Related Topics:
Operation - Infinitesimal

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:lim_{h o 0}rac{f(x+h) - f(x)}{h}.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: f'(x) quad

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

pronounced "f prime of x" or "f dash of x"

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: rac{d}{dx} f (x)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: rac{df}{dx}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

pronounced "d f by d x" or "d f d x"

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: D_x f quad

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

pronounced "d sub x of f"

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:dot{x}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

pronounced "x dot".

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 x, then there is no tangent line and the function is therefore not differentiable at x; however, even if a function is continuous at x, it may not be differentiable there. In other words, differentiability implies continuity, but not vice versa. One famous example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function.

Related Topics:
Interval - Continuous - Weierstrass function

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~