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3.5.3 Derivatives of Unknown Functions

  • Differentiating a known function gives an explicit result.
  • In[1]:= D[Log[x]^2, x]


  • Differentiating an unknown function f gives a result in terms of f'.
  • In[2]:= D[f[x]^2, x]


  • Mathematica applies the chain rule for differentiation, and leaves the result in terms of f'.
  • In[3]:= D[x f[x^2], x]


  • Differentiating again gives a result in terms of f, f' and f''.
  • In[4]:= D[%, x]


  • When a function has more than one argument, superscripts are used to indicate how many times each argument is being differentiated.
  • In[5]:= D[g[x^2, y^2], x]


  • This represents . Mathematica

    assumes that the order in which derivatives are taken with respect to different variables is irrelevant.
  • In[6]:= D[g[x, y], x, x, y]


  • You can find the value of the derivative when by replacing x with 0

  • In[7]:= % /. x->0


    Output forms for derivatives of unknown functions.