Mathematica 9 is now available
THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT.
SEE THE DOCUMENTATION CENTER FOR THE LATEST INFORMATION.
Wolfram Mathematica Documentation Center
DOCUMENTATION CENTER SEARCH
New to Mathematica? Find your learning path »
Mathematica > Mathematics and Algorithms > Calculus > D () >
BUILT-IN MATHEMATICA SYMBOLTutorials »|See Also »|More About »

D

D
gives the partial derivative .
D
gives the multiple derivative .
D
differentiates f successively with respect to .
D
for a scalar f gives the vector derivative .
D
gives a tensor derivative.
MORE INFORMATION
  • D can be input as . The character is entered as Esc pd Esc or . The variable x is entered as a subscript.
  • All quantities that do not explicitly depend on the variables given are taken to have zero partial derivative.
  • D[f, var1, ..., NonConstants->{u1, ...}] specifies that every implicitly depends on every , so that they do not have zero partial derivative.
  • D threads over lists that appear in f.
  • D effectively threads D over each element of list.
  • D is equivalent to D where {list} is repeated n times. If f is a scalar, and list has depth 1, then the result is a tensor of rank n, as in the n^(th) term of the multivariate Taylor series of f.
  • D is normally equivalent to First[Outer[D, {f}, list1, list2, ...]].
  • If f is a list, then D effectively threads first over each element of f, and then over each element of list. The result is an array with dimensions Join[Dimensions[f], Dimensions[list]].
  • Numerical approximations to derivatives can be found using N.
  • D uses the chain rule to simplify derivatives of unknown functions.
  • D can be input as . The character , entered as Esc , Esc, can be used instead of an ordinary comma. It does not display, but is still interpreted just like a comma.
EXAMPLESCLOSE ALL
Basic Examples   (7)
Derivative with respect to :
Fourth derivative with respect to :
Derivative with respect to and :
Derivative involving a symbolic function :
Vector derivative (gradient vector):
Second-order derivative tensor:
Evaluate derivatives numerically:
Enter using Esc pd Esc, and subscripts using Control+_:
Derivative with respect to :
In[1]:=
Click for copyable input
Out[1]=
 
Fourth derivative with respect to :
In[1]:=
Click for copyable input
Out[1]=
 
Derivative with respect to and :
In[1]:=
Click for copyable input
Out[1]=
 
Derivative involving a symbolic function :
In[1]:=
Click for copyable input
Out[1]=
 
Vector derivative (gradient vector):
In[1]:=
Click for copyable input
Out[1]=
Second-order derivative tensor:
In[2]:=
Click for copyable input
Out[2]=
 
Evaluate derivatives numerically:
In[1]:=
Click for copyable input
Out[1]=
 
Enter using Esc pd Esc, and subscripts using Control+_:
In[1]:=
Click for copyable input
Out[1]=
Generalizations & Extensions   (1)
Differentiate with respect to different formal variables:
Options   (1)
Differentiate with y considered as depending on x:
Solve for the derivative of y to effect implicit differentiation:
Applications   (5)
Find the turning points on a plane curve:
Perform the change of variable in an integral:
Find the curvature of a circular helix with radius r and pitch c:
Compute the coefficients of a power series:
Construct the differential equation satisfied by an implicit function :
Possible Issues   (2)
Results may not immediately be given in the simplest possible form:
Functions given in different forms can yield the same derivatives:
New in 1 | Last modified in 7
Ask a question about this page  |  Suggest an improvement  |  Leave a message for the team
Format:   HTML  |  CDF