|Dt[f]||total differential df|
|Dt[f,x]||total derivative |
|Dt[f,x,y,...]||multiple total derivative |
|total derivative with ci constant (i.e., dci=0)|
|SetAttributes[c,Constant]||define c to be a constant in all cases|
Total differentiation operations.
When you find the derivative of some expression f
with respect to x
, you are effectively finding out how fast f
changes as you vary x
. Often f
will depend not only on x
, but also on other variables, say y
. The results that you get then depend on how you assume that y
vary as you change x
There are two common cases. Either y
are assumed to stay fixed when x
changes, or they are allowed to vary with x
. In a standard partial derivative
, all variables other than x
are assumed fixed. On the other hand, in the total derivative
, all variables are allowed to change with x
, D[f, x]
gives a partial derivative, with all other variables assumed independent of x
. Dt[f, x]
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 x
This gives the total derivative
. Now y
is assumed to depend on x
You can make a replacement for
You can also make an explicit definition for
. You need to use y/:
to make sure that the definition is associated with y
With this definition made, Dt
as independent of x
This removes your definition for the derivative of y
This takes the total derivative, with z
This specifies that c
is a constant under differentiation.
The variable c
is taken as a constant.
The function c
is also assumed to be a constant.
This gives the total differential d (x2+cy2)
You can make replacements and assignments for total differentials.