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.)

Notations for differentiation

The simplest notation for differentiation that is in current use is due to Lagrange and uses the prime, ′. To take derivatives of f(x) at the point a, we write:

Related Topics:
Lagrange - Prime

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:f ′(a) for the first derivative,

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:f ″(a) for the second derivative,

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:f ″′(a) for the third derivative and then

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:f(n)(a) for the nth derivative (n > 3).

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For the function whose value at each x is the derivative of f(x), we write f ′(x). Similarly, for the second derivative of f we write f ″(x), and so on.

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The other common notation is Leibniz's notation for differentiation which is named after Leibniz. For the function whose value at x is the derivative of f at x, we write:

Related Topics:
Leibniz's notation for differentiation - Leibniz

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:rac{dleft(f(x) ight)}{dx}.

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We can write the derivative of f at the point a in two different ways:

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:rac{dleft(f(x) ight)}{dx}left.{!!rac{}{}} ight|_{x=a} = left(rac{dleft(f(x) ight)}{dx} ight)(a).

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If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as:

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:rac{dy}{dx}.

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Higher derivatives are expressed as

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:rac{d^nleft(f(x) ight)}{dx^n} or rac{d^ny}{dx^n}

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for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is:

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

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which we can loosely write as:

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:left(rac{d}{dx} ight)^3 left(f(x) ight) =

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rac{d^3}{left(dx ight)^3} left(f(x) ight).

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Dropping brackets gives the notation above.

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Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember, because the "du" terms appear symbolically to cancel:

Related Topics:
Partial differentiation - Chain rule

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:rac{dy}{dx} = rac{dy}{du} cdot rac{du}{dx}.

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(In the popular formulation of calculus in terms of limits, the "du" terms cannot literally cancel, because on their own they are undefined; they are only defined when used together to express a derivative. In nonstandard analysis, however, they can be viewed as infinitesimal numbers that cancel.)

Related Topics:
Nonstandard analysis - Infinitesimal

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Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the function name:

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:dot{x} = rac{dx}{dt} = x'(t)

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:ddot{x} = x(t)

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and so on.

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Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration, and in ODE theory. It is usually only used for first and second derivatives.

Related Topics:
Mechanics - ODE

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Euler's notation uses a differential operator denoted as D which when prefixed to a function f gives the derivative of f:

Related Topics:
Euler - Differential operator

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: Df = f' .

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Higher derivatives are obtained by repeated application of the D operator, which can be abbreviated by exponentiating a single D, e.g.

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: D D D f = D^3 f = f.

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Euler's notation is useful for stating and solving linear differential equations.

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