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Calculus and Analysis > Calculus > Differential Calculus v
MathWorld Contributors > Derwent v

Derivative

COMMENT On this Page EXPLORE THIS TOPIC IN the MathWorld ClassroomDOWNLOAD Mathematica Notebook

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 a variable x is denoted either f^'(x) or
(df)/(dx), (1)

often written in-line as df/dx. When derivatives are taken with respect to time, they are often denoted using Newton's Eric Weisstein's World of Biography overdot notation for fluxions,
(dx)/(dt)==x^.. (2)

When a derivative is taken n times, the notation f^((n))(x) or
(d^nf)/(dx^n) (3)

is used, with
x^.,x^..,x^..., (4)

etc., the corresponding fluxion notation. When a function f(x,y,...) depends on more than one variable, a partial derivative
(partialf)/(partialx),(partial^2f)/(partialxpartialy), etc. (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
f^'(x)=lim_(h->0)(f(x+h)-f(x))/h, (6)

but may also be calculated more symmetrically as
f^'(x)==lim_(h->0)(f(x+h)-f(x-h))/(2h), (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 the the existence of complex derivatives, variables, or functions. For example, textbook examples to the contrary, the "derivative" (read: complex derivative) d|z|/dz of the absolute value function |z| 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 x==0 as
(d|x|)/(dx)=={-1 for x<0; undefined for x==0; 1 for x>0. (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), d|x|/dx correctly returns unevaluated by such software.

If the first derivative exists, the second derivative may be defined as
f^('')(x)=lim_(h->0)(f^'(x+h)-f^'(x))/h (9)

and calculated more symmetrically as
f^('')(x)==lim_(h->0)(f(x+2h)-2f(x+h)+f(x))/(h^2), (10)

again provided the second derivative is known to exist.

Note that in order for the limit to exist, both lim_(h->0^+) and lim_(h->0^-) 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 Eric Weisstein's World of Biography 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.
d/(dx)x^n = nx^(n-1) (11)
d/(dx)lnx = 1/x (12)
d/(dx)sinx = cosx (13)
d/(dx)cosx = -sinx (14)
d/(dx)tanx = sec^2x (15)
d/(dx)cscx = -cscxcotx (16)
d/(dx)secx = secxtanx (17)
d/(dx)cotx = -csc^2x (18)
d/(dx)e^x = e^x (19)
d/(dx)a^x = (lna)a^x (20)
d/(dx)sin^(-1)x = 1/(sqrt(1-x^2)) (21)
d/(dx)cos^(-1)x = -1/(sqrt(1-x^2)) (22)
d/(dx)tan^(-1)x = 1/(1+x^2) (23)
d/(dx)cot^(-1)x = -1/(1+x^2) (24)
d/(dx)sec^(-1)x = 1/(xsqrt(x^2-1)) (25)
d/(dx)csc^(-1)x = -1/(xsqrt(x^2-1)) (26)
d/(dx)sinhx = coshx (27)
d/(dx)coshx = sinhx (28)
d/(dx)tanhx = sech^2x (29)
d/(dx)cothx = -csch^2x (30)
d/(dx)sechx = -sechxtanhx (31)
d/(dx)cschx = -cschxcothx (32)
d/(dx)snx = cnxdnx (33)
d/(dx)cnx = -snxdnx (34)
d/(dx)dnx = -k^2snxcnx. (35)

where sn(x)=sn(x,k), cn(x)=cn(x,k), 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
[f(x)+...+h(x)]^'==f^'(x)+...+h^'(x). (36)

In addition, if c is a constant,
d/(dx)[cf(x)]==cf^'(x). (37)

The product rule for differentiation states
d/(dx)[f(x)g(x)]==f(x)g^'(x)+f^'(x)g(x), (38)

where f^' denotes the derivative of f with respect to x. This derivative rule can be applied iteratively to yield derivative rules for products of three or more functions, for example,
[fgh]^' = (fg)h^'+(fg)^'h==fgh^'+(fg^'+f^'g)h (39)
= f^'gh+fg^'h+fgh^'. (40)

The quotient rule for derivatives states that
d/(dx)[(f(x))/(g(x))]==(g(x)f^'(x)-f(x)g^'(x))/([g(x)]^2) (41)

while the power rule gives
d/(dx)(x^n)==nx^(n-1). (42)

Other very important rule for computing derivatives is the chain rule, which states that for y==y(u),
(dy)/(dx)==(dy)/(du).(du)/(dx), (43)

or more generally, for z==z(x(t),y(t))
(dz)/(dt)==(partialz)/(partialx)(dx)/(dt)+(partialz)/(partialy)(dy)/(dt), (44)

where partialz/partialx denotes a partial derivative.

Miscellaneous other derivative identities include
(dy)/(dx)==((dy)/(dt))/((dx)/(dt)) (45)
(dy)/(dx)==1/((dx)/(dy)). (46)

If F(x,y)==C, where C is a constant, then
dF==(partialF)/(partialy)dy+(partialF)/(partialx)dx==0, (47)

so
(dy)/(dx)==-((partialF)/(partialx))/((partialF)/(partialy)). (48)

Derivative identities of inverse functions include
(dx)/(dy) = 1/((dy)/(dx)) (49)
(d^2x)/(dy^2) = -(d^2y)/(dx^2)((dy)/(dx))^(-3) (50)
(d^3x)/(dy^3) = [3((d^2y)/(dx^2))^2-(d^3y)/(dx^3)(dy)/(dx)]((dy)/(dx))^(-5). (51)

A vector derivative of a vector function
X(t)=[x_1(t); x_2(t); |; x_k(t)] (52)

can be defined by
(dX)/(dt)==[(dx_1)/(dt); (dx_2)/(dt); |; (dx_k)/(dt)]. (53)

The nth derivatives of x^nf(x) for n==1, 2, ... are
d/(dx)[xf(x)] = f(x)+xf^'(x) (54)
(d^2)/(dx^2)[x^2f(x)] = 2f(x)+4xf^'(x)+x^2f^('')(x) (55)
(d^3)/(dx^3)[x^3f(x)] = 6f(x)+18xf^'(x)+9x^2f^('')(x)+x^3f^(''')(x). (56)

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 L_n(x).

Faá di Bruno's formula gives an explicit formula for the nth derivative of the composition f(g(t)).

SEE ALSO: Blancmange Function, Calculus, Carathéodory Derivative, Cauchy-Riemann Equations, Chain Rule, Comma Derivative, Complex Derivative, Complex Differentiable, Convective Derivative, Covariant Derivative, Definite Integral, Differentiable, Differential Calculus, Differentiation, Directional Derivative, Euler-Lagrange Derivative, Faá di Bruno's Formula, Finite Difference, Fluxion, Fractional Calculus, Fréchet Derivative, Functional Derivative, Implicit Differentiation, Indefinite Integral, Integral, Lagrangian Derivative, Lie Derivative, Logarithmic Derivative, Numerical Differentiation, Pincherle Derivative, Power Rule, Product Rule, q-Derivative, Quotient Rule, Schwarzian Derivative, Semicolon Derivative, Total Derivative, Weierstrass Function. [Pages Linking Here]


Derivatives are fun.