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Defining Derivatives

You can define the derivative in Mathematica of a function f of one argument simply by an assignment like f'[x_]=fp[x].
This defines the derivative of f (x) to be fp (x). In this case, you could have used = instead of :=.
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The rule for f'[x_] is used to evaluate this derivative.
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Differentiating again gives derivatives of fp.
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This defines a value for the derivative of g at the origin.
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The value for g'[0] is used.
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This defines the second derivative of g, with any argument.
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The value defined for the second derivative is used.
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To define derivatives of functions with several arguments, you have to use the general representation of derivatives in Mathematica.
f'[x_]:=rhsdefine the first derivative of f
Derivative[n][f][x_]:=rhsdefine the nth derivative of f
Derivative[m,n,...][g][x_,_,...]:=rhs
define derivatives of g with respect to various arguments

Defining derivatives.

This defines the second derivative of g with respect to its second argument.
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This uses the definition just given.
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