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3.5.1 Differentiation
| D[f, x] | partial derivative  | | D[f, x, y, ... ] | multiple derivative  | | D[f, {x, n}] | n derivative  |
D[f, x, NonConstants -> { ,  , ... }]
| |  with the  taken to depend on  |
Partial differentiation operations. This gives  . | |
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| This gives the third derivative. | |
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| You can differentiate with respect to any expression that does not involve explicit mathematical operations. | |
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D[ x[1]^2 + x[2]^2, x[1] ]
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| D does partial differentiation. It assumes here that y is independent of x. | |
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| If y does in fact depend on x, you can use the explicit functional form y[x]. Section 3.5.4 describes how objects like y'[x] work. | |
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D[x^2 + y[x]^2, x]
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Instead of giving an explicit function y[x], you can tell D that y implicitly depends on x. D[y, x, NonConstants->{y}] then represents  , with y implicitly depending on x. | |
In[6]:=
D[x^2 + y^2, x, NonConstants -> {y}]
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D[f, {{ ,  , ... }}] | the gradient of a scalar function f  | D[f, {{ ,  , ... }, 2}] | the Hessian matrix for f | D[f, {{ ,  , ... }, n}] | the   order Taylor series coefficient | D[{ ,  , ... }, {{ ,  , ... }}] | the Jacobian for a vector function f |
Vector derivatives. This gives the gradient of the function  . | |
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D[x^2 + y^2, {{x, y}}]
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| This gives the Hessian. | |
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D[x^2 + y^2, {{x, y}, 2}]
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| This gives the Jacobian for a vector function. | |
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D[{x^2 + y^2, x y}, {{x, y}}]
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