This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)
 Documentation / Mathematica / The Mathematica Book / Practical Introduction / Symbolic Mathematics  /

1.5.2 Differentiation

  • Here is the derivative of with respect to

  • In[1]:= D[ x^n, x ]


  • Mathematica knows the derivatives of all the standard mathematical functions.
  • In[2]:= D[ ArcTan[x], x ]


  • This differentiates three times with respect to x.
  • In[3]:= D[ x^n, {x, 3} ]


    The function D[x^n,x] really gives a partial derivative, in which n is assumed not to depend on x. Mathematica has another function, called Dt, which finds total derivatives, in which all variables are assumed to be related. In mathematical notation, D[f,x] is like , while Dt[

    f,x] is like . You can think of Dt

    as standing for "derivative total".

  • Dt gives a total derivative, which assumes that n can depend on x. Dt[n,x] stands for

  • In[4]:= Dt[ x^n, x ]


  • This gives the total differential . Dt[x] is the differential

  • In[5]:= Dt[ x^n ]


    Some differentiation functions.

    As well as treating variables like symbolically, you can also treat functions in Mathematica symbolically. Thus, for example, you can find formulas for derivatives of f[x], without specifying any explicit form for the function f


  • Mathematica does not know how to differentiate f, so it gives you back a symbolic result in terms of f'.
  • In[6]:= D[ f[x], x ]


  • Mathematica uses the chain rule to simplify derivatives.
  • In[7]:= D[ 2 x f[x^2], x ]