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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]= 
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