This is documentation for Mathematica 4, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)
Wolfram Research, Inc.

Basic OperationsIntegration

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 ]


Basic OperationsIntegration