 |
|
The derivative of a function
represents an infinitesimal change in the function with respect to
whatever parameters it may have. The "simple" derivative of a
function f with respect to x is denoted either  or
 |
(1) |
(and often written in-line as  ). When derivatives are taken with respect to time, they
are often denoted using Newton's
 overdot
notation for fluxions,
 |
(2) |
When a derivative is taken n times, the notation  or
 |
(3) |
is used, with
 |
(4) |
the corresponding fluxion
notation. When a function  depends on more than one variable, a partial
derivative
 |
(5) |
can be used to specify the derivative with respect to one or more
variables.
The derivative of a function f(x) with respect to
the variable x is defined as
 |
(6) |
Note that in order for the limit to exist, both  and  must exist and be equal, so the function must
be continuous. However, continuity is a necessary but
not sufficient
condition for differentiability. Since some discontinuous
functions can be integrated, in a sense there are "more" functions
which can be integrated than differentiated. In a letter to
Stieltjes, Hermite
wrote, "I recoil with dismay and horror at this lamentable plague of
functions which do not have derivatives."
A three-dimensional generalization of the derivative to an
arbitrary direction is known as the directional
derivative. In general, derivatives are mathematical objects
which exist between smooth functions on manifolds. In this
formalism, derivatives are usually assembled into "tangent
maps."
Performing numerical
differentiation is in many ways more difficult than numerical
integration. This is because while numerical
integration requires only good continuity properties of the
function being integration, numerical
differentiation requires more complicated properties such as
Lipschitz classes.
Simple derivatives of some simple functions follow.
 |
(7) |
 |
(8) |
 |
(9) |
 |
(10) |
 |
(11) |
 |
(12) |
 |
(13) |
 |
(14) |
 |
(15) |
 |
(16) |
 |
(17) |
 |
(18) |
 |
(19) |
 |
(20) |
 |
(21) |
 |
(22) |
 |
(23) |
 |
(24) |
 |
(25) |
 |
(26) |
 |
(27) |
 |
(28) |
 |
(29) |
 |
(30) |
 |
(31) |
where  ,  , etc. are Jacobi
elliptic functions, and the product
rule and quotient
rule have been used extensively to expand the derivatives.
There are a number of important rules for computing derivatives
of certain combinations of functions. Derivatives of sums are equal
to the sum of derivatives so that
 |
(32) |
In addition, if c is a constant,
 |
(33) |
The product
rule for differentiation states
 |
(34) |
where  denotes the derivative
of f with respect to x. This derivative rule can be
applied iteratively to yield derivate rules for products of three or
more functions, for example,
The quotient
rule for derivatives states that
 |
(36) |
while the power rule
gives
 |
(37) |
Other very important rule for computing derivatives is the chain rule,
which states that
 |
(38) |
or more generally,
 |
(39) |
were  denotes a partial
derivative.
Miscellaneous other derivative identities include
 |
(40) |
 |
(41) |
If  , where C is a constant, then
 |
(42) |
so
 |
(43) |
A vector derivative of a vector function
 |
(44) |
can be defined by
 |
(45) |
The nth derivatives of  for n = 1, 2, ... are
The nth row of the triangle of coefficients 1; 1,
1; 2, 4, 1; 6, 18, 9, 1; ... (Sloane's A021009)
is given by the absolute values of the coefficients of the Laguerre
polynomial  .
Faá di
Bruno's formula gives an explicit formula for the nth
derivative of the composition
 .
 Blancmange
Function, Carathéodory
Derivative, Chain Rule,
Comma
Derivative, Convective
Derivative, Covariant
Derivative, Differentiation,
Directional
Derivative, Euler-Lagrange
Derivative, Faá di
Bruno's Formula, Fluxion, Fractional
Calculus, Fréchet
Derivative, Functional
Derivative, Implicit
Differentiation, Lagrangian
Derivative, Lie
Derivative, Logarithmic
Derivative, Numerical
Differentiation, Pincherle
Derivative, Power Rule,
Product
Rule, q-Series,
Quotient
Rule, Schwarzian
Derivative, Semicolon
Derivative, Weierstrass
Function
References
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing. New York: Dover, p. 11, 1972.
Anton, H. Calculus:
A New Horizon, 6th ed. New York: Wiley, 1999.
calc101.com. "Step-by-Step Differentiation." http://www.calc101.com/webMathematica/MSP/Calc101/WalkD.
Beyer, W. H. "Derivatives." CRC
Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC
Press, pp. 229-232, 1987.
Griewank, A. Principles
and Techniques of Algorithmic Differentiation. Philadelphia,
PA: SIAM, 2000.
Sloane, N. J. A. Sequences A021009
in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.
Eric W. Weisstein
© 1999 CRC Press LLC, ©
1999-2004 Wolfram Research,
Inc.
|
