Defining Derivatives

You can define the derivative in Mathematica of a function of one argument simply by an assignment like .

This defines the derivative of to be . In this case, you could have used instead of .
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The rule for is used to evaluate this derivative.
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Differentiating again gives derivatives of .
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This defines a value for the derivative of at the origin.
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The value for is used.
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This defines the second derivative of , 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 n^(th) 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 with respect to its second argument.
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This uses the definition just given.
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