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1.5.2 Differentiation

Here is the derivative of with respect to .

In[1]:= D[ x^n, x ]

Out[1]=

Mathematica knows the derivatives of all the standard mathematical functions.

In[2]:= D[ ArcTan[x], x ]

Out[2]=

This differentiates three times with respect to x.

In[3]:= D[ x^n, {x, 3} ]

Out[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 ]

Out[4]=

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

In[5]:= Dt[ x^n ]

Out[5]=

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 ]

Out[6]=

Mathematica uses the chain rule to simplify derivatives.

In[7]:= D[ 2 x f[x^2], x ]

Out[7]=