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Advanced Topic: The Representation of DerivativesIndefinite Integrals

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]=

Advanced Topic: The Representation of DerivativesIndefinite Integrals