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D (PartialD)

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D[f, x]
gives the partial derivative partialf/partialx.
D[f, {x, n}]
gives the multiple derivative partial^nf/partialx^n.
D[f, x, y, ...]
differentiates f successively with respect to x,y,....
D[f, {{x1, x2, ...}}]
for a scalar f gives the vector derivative (partialf/partialx_1,partialf/partialx_2,...).
D[f, {array}]
gives a tensor derivative.
  • D[f, x] can be input as partial_xf. The character partial is entered as Esc pd Esc or \[PartialD]. 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 u_i implicitly depends on every varj, so that they do not have zero partial derivative.
  • D[f, ...] threads over lists that appear in f.
  • D[f, {list}] effectively threads D over each element of list.
  • D[f, {list, n}] is equivalent to D[f, {list}, {list}, ...] 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[f, {list1}, {list2}, ...] is normally equivalent to First[Outer[D, {f}, list1, list2, ...]].
  • If f is a list, then D[f, {list}] 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[f, x, y] can be input as partial_(x,y)f. The character \[InvisibleComma], 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 x:
4^(th) derivative with respect to x:
Derivative with respect to x and y:
Derivative involving a symbolic function f:
Vector derivative (gradient vector):
2^(nd)-order derivative tensor:
Evaluate derivatives numerically:
Enter partial using Esc pd Esc, and subscripts using Control+_:
Derivative with respect to x:
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4^(th) derivative with respect to x:
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Derivative with respect to x and y:
In[1]:=
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Out[1]=
 
Derivative involving a symbolic function f:
In[1]:=
Click for copyable input
Out[1]=
 
Vector derivative (gradient vector):
In[1]:=
Click for copyable input
Out[1]=
2^(nd)-order derivative tensor:
In[2]:=
Click for copyable input
Out[2]=
 
Evaluate derivatives numerically:
In[1]:=
Click for copyable input
Out[1]=
 
Enter partial 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 t = x^2 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 y[x]:
Possible Issues   (2)
Results may not immediately be given in the simplest possible form:
Functions given in different forms can yield the same derivatives:
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