D

D[f,x]

gives the partial derivative .

D[f,{x,n}]

gives the multiple derivative .

D[f,x,y,…]

gives the partial derivative .

D[f,{x,n},{y,m},…]

gives the multiple partial derivative .

D[f,{{x1,x2,…}}]

for a scalar f gives the vector derivative .

D[f,{array}]

gives an array derivative.

Details and Options

  • D is also known as derivative for univariate functions.
  • By using the character ∂, entered as pd or \[PartialD], with subscripts, derivatives can be entered as follows:
  • D[f,x]∂xf
    D[f,{x,n}]∂{x,n}f
    D[f,x,y]∂x,yf
    D[f,{{x,y}}]∂{{x,y}}f
  • The comma can be made invisible by using the character \[InvisibleComma] or ,.
  • The partial derivative D[f[x],x] is defined as , and higher derivatives D[f[x,y],x,y] are defined recursively as etc.
  • The order of derivatives n and m can be symbolic and they are assumed to be positive integers.
  • The derivative D[f[x],{x,n}] for a symbolic f is represented as Derivative[n][f][x].
  • For some functions f, Derivative[n][f][x] may not be known, but can be approximated by applying N. »
  • New derivative rules can be added by adding values to Derivative[n][f][x]. »
  • For lists, D[{f1,f2,…},x] is equivalent to {D[f1,x],D[f2,x],…} recursively. »
  • D[f,{array}] effectively threads D over each element of array.
  • D[f,{array,n}] is equivalent to D[f,{array},{array},…], where {array} is repeated n times.
  • D[f,{array1},{array2},…] is normally equivalent to First[Outer[D,{f},array1,array2,…]]. »
  • Common array derivatives include:
  • D[f,{{x1,x2,…}}]gradient{D[f,x1],D[f,x2],…}
    D[f,{{x1,x2,…},2}]Hessian{{D[f,x1,x1],D[f,x1,x2],…},{D[f,x2,x1],D[f,x2,x2],…},…}
    D[{f1,f2,…},{{x1,x2,…}}]Jacobian{{D[f1,x1],D[f1,x2],…},
    {D[f2,x1],D[f2,x2],…},…}
  • If f is a scalar and x={x1,…}, then the multivariate Taylor series at x0={x01,…} is given by:
  • ,
  • where fi=D[f,{x,i}]/.{x1x01,…} is an array with tensor rank . »
  • If f and x are both arrays, then D[f,{x}] effectively threads first over each element of f, and then over each element of x. The result is an array with dimensions Join[Dimensions[f],Dimensions[x]]. »
  • D can formally differentiate operators such as integrals and sums, taking into account scoped variables as well as the structure of the particular operator.
  • Examples of operator derivatives include:
  • is not scoped by the integral
    is scoped by the integral
    is not scoped by the integral transform
    is scoped by by the integral transform
  • All expressions that do not explicitly depend on the variables given are taken to have zero partial derivative.
  • The setting NonConstants{u1,…} specifies that ui depends on all variables x, y, etc. and does not have zero partial derivative. »

Examples

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Basic Examples  (7)

Derivative with respect to x:

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Fourth derivative with respect to x:

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Derivative of order n with respect to x:

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Derivative with respect to x and y:

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Derivative involving a symbolic function f:

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Evaluate derivatives numerically:

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Enter ∂ using pd, and subscripts using :

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Scope  (82)

Options  (1)

Applications  (41)

Properties & Relations  (21)

Possible Issues  (4)

Interactive Examples  (2)

Neat Examples  (2)

See Also

Dt  Derivative  DifferenceQuotient  Minimize  Maximize  DSolve  NDSolve  NDEigensystem  NDEigenvalues  Integrate  Grad  Div  Curl  Laplacian  DifferenceDelta  CoefficientArrays

Tutorials

Introduced in 1988
(1.0)
| Updated in 2017
(11.1)