3.5.5 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 to be . In this case, you could have used = instead of :=.
The rule for f'[x_] is used to evaluate this derivative.
Differentiating again gives derivatives of .
This defines a value for the derivative of at the origin.
The value for g' is used.
This defines the second derivative of g, with any argument.
The value defined for the second derivative is used.
To define derivatives of functions with several arguments, you have to use the general representation of derivatives in Mathematica.
This defines the second derivative of g with respect to its second argument.
This uses the definition just given.