Total Derivatives
| Dt[f] | total differential  |
| Dt[f,x] | total derivative  |
| Dt[f,x,y,...] | multiple total derivative  |
| Dt[f,x,Constants->{c1,c2,...}] | total derivative with  constant (i.e.,  ) |
| y/:Dt[y,x]=0 | set  |
| SetAttributes[c,Constant] | define c to be a constant in all cases |
Total differentiation operations.
When you find the derivative of some expression
with respect to
, you are effectively finding out how fast
changes as you vary
. Often
will depend not only on
, but also on other variables, say
and
. The results that you get then depend on how you assume that
and
vary as you change
.
There are two common cases. Either
and
are assumed to stay fixed when
changes, or they are allowed to vary with
. In a standard
partial derivative 
, all variables other than
are assumed fixed. On the other hand, in the
total derivative 
, all variables are allowed to change with
.
In
Mathematica,
D
gives a partial derivative, with all other variables assumed independent of
x.
Dt
gives a total derivative, in which all variables are assumed to depend on
x. In both cases, you can add an argument to give more information on dependencies.
This gives the
partial derivative 
.
is assumed to be independent of
.
| Out[1]= |  |
This gives the
total derivative 
. Now
is assumed to depend on
.
| Out[2]= |  |
You can make a replacement for
.
| Out[3]= |  |
You can also make an explicit definition for
. You need to use
to make sure that the definition is associated with
.
| Out[4]= |  |
With this definition made,
Dt treats
as independent of
.
| Out[5]= |  |
This removes your definition for the derivative of
.
This takes the total derivative, with
held fixed.
| Out[7]= |  |
This specifies that
is a constant under differentiation.
The variable
is taken as a constant.
| Out[9]= |  |
The
function 
is also assumed to be a constant.
| Out[10]= |  |
This gives the total differential
.
| Out[11]= |  |
You can make replacements and assignments for total differentials.
| Out[12]= |  |
