The derivative of a function represents
an infinitesimal change in the function with respect to one of its variables.
The "simple" derivative of a function with respect to
a variable is denoted either or
|
(1)
|
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)
|
The "d-ism" of Leibnitz's eventually
won the notation battle against the "dotage" of Newton's fluxion notation
(P. Ion, pers. comm., Aug. 18, 2006).
When a derivative is taken times, the notation or
|
(3)
|
is used, with
|
(4)
|
etc., 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 with respect
to the variable is defined as
|
(6)
|
but may also be calculated more symmetrically as
|
(7)
|
provided the derivative is known to exist.
It should be noted that the above definitions refer to "real" derivatives, i.e., derivatives which are restricted to directions along the real axis. However, this restriction is artificial, and derivatives
are most naturally defined in the complex
plane, where they are sometimes explicitly referred to as complex derivatives. In order for complex derivatives to exist,
the same result must be obtained for derivatives taken in any direction in
the complex plane. Somewhat surprisingly,
almost all of the important functions in mathematics satisfy this property, which
is equivalent to saying that they satisfy the Cauchy-Riemann equations.
These considerations can lead to confusion for students because elementary calculus texts commonly consider only "real" derivatives, never alluding to the
existence of complex derivatives, variables, or functions. For example, textbook
examples to the contrary, the "derivative" (read: complex derivative) of the absolute value function does not exist
because at every point in the complex
plane, the value of the derivative depends on the direction in which the derivative
is taken (so the Cauchy-Riemann
equations cannot and do not hold). However, the real derivative (i.e.,
restricting the derivative to directions along the real
axis) can be defined for points other than as
|
(8)
|
As a result of the fact that computer algebra programs such as Mathematica generically deal with complex variables
(i.e., the definition of derivative always means complex derivative), correctly returns unevaluated by such software.
If the first derivative exists, the second derivative may be defined as
|
(9)
|
and calculated more symmetrically as
|
(10)
|
again provided the second derivative is known to exist.
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 integrated,
numerical differentiation
requires more complicated properties such as Lipschitz classes.
Simple derivatives of some simple functions follow:
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
|
(36)
|
In addition, if is a constant,
|
(37)
|
The product rule for differentiation
states
|
(38)
|
where denotes the derivative of with respect to
. This derivative rule can be applied iteratively
to yield derivative rules for products of three or more functions, for example,
The quotient rule for derivatives
states that
|
(42)
|
while the power rule gives
|
(43)
|
Other very important rule for computing derivatives is the chain rule, which states that for ,
|
(44)
|
or more generally, for
|
(45)
|
where denotes a partial derivative.
Miscellaneous other derivative identities include
|
(46)
|
|
(47)
|
If , where is a constant,
then
|
(48)
|
so
|
(49)
|
Derivative identities of inverse functions include
A vector derivative of a vector function
|
(53)
|
can be defined by
|
(54)
|
The th derivatives of for , 2, ... are
The th 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 .
Faà di Bruno's formula gives an explicit formula for the th derivative of
the composition .
The June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007) featured the following derivative as a "hard" exam
problem intended for a remedial math class but accidentally handed out to the normal
class:
|
(58)
|
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.
Amend, B. Camp FoxTrot. Kansas City, MO: Andrews McMeel, p. 19,
1998.
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.
Mitchell, C. W. Jr. In "Media Clips" (Ed. M. Cibes and J. Greenwood). Math. Teacher 100, 339, Dec. 2006/Jan. 2007. Sloane, N. J. A.
Sequence A021009
in "The On-Line Encyclopedia of Integer Sequences."
|