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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 :=.
In[1]:= f'[x_] := fp[x]
The rule for f'[x_] is used to evaluate this derivative.
In[2]:= D[f[x^2], x]
Out[2]= 
Differentiating again gives derivatives of  .
In[3]:= D[%, x]
Out[3]= 
This defines a value for the derivative of  at the origin.
In[4]:= g'[0] = g0
Out[4]= 
The value for g'[0] is used.
In[5]:= D[g[x]^2, x] /. x->0
Out[5]= 
This defines the second derivative of g, with any argument.
In[6]:= g''[x_] = gpp[x]
Out[6]= 
The value defined for the second derivative is used.
In[7]:= D[g[x]^2, {x, 2}]
Out[7]= 
To define derivatives of functions with several arguments, you have to use the general representation of derivatives in Mathematica.
Defining derivatives.
This defines the second derivative of g with respect to its second argument.
In[8]:= Derivative[0, 2][g][x_, y_] := g2p[x, y]
This uses the definition just given.
In[9]:= D[g[a^2, x^2], x, x]
Out[9]= 
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