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D ()

D[f, x]
gives the partial derivative ∂f/∂x.
D[f, {x, n}]
gives the multiple derivative ∂^nf/∂x^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 (∂f/∂x_1,∂f/∂x_2,…).
  • D[f, x] can be input as ∂_xf. The character ∂ 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, {list}] 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, a 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, ...]].
  • 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 ∂_(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:
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4^(th) derivative 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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Vector derivative (gradient vector):
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2^(nd)-order derivative tensor:
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Evaluate derivatives numerically:
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Enter ∂ using Esc pd Esc, and subscripts using Control+_:
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