## PartialDerivative

Makes an object a partial derivative.

Makes an object a partial derivative, i.e. a derivative which
commutes. The object on which it acts has to be a non-sub/superscript
child, while all the sub- or superscript child nodes are interpreted
to be the variables with respect to which the derivative is taken.\partial{#}::PartialDerivative.
A_{\mu}::Depends(\partial{#}).
ex:= \partial_{\nu}{A_{\mu} B_{\rho}};

product_rule(_);

\(\displaystyle{}\partial_{\nu}{A_{\mu}} B_{\rho}+A_{\mu} \partial_{\nu}{B_{\rho}}\)

unwrap(_);

\(\displaystyle{}\partial_{\nu}{A_{\mu}} B_{\rho}\)

\partial_{\nu}(A_{\mu}) B_{\rho}

Note that derivative objects do not necessarily need to have a sub- or
superscript child, they can be abstract derivatives as in

D{#}::PartialDerivative.
ex:= D(c d e);

\(\displaystyle{}D\left(c d e\right)\)

D(c d e)

\partial(f g)

product_rule(_);

\(\displaystyle{}D{c} d e+c D{d} e+c d D{e}\)

D(c) d e + c D(d) e + c d D(e)

g \partial(f)

If you want to write a derivative with respect to a coordinate (instead
of with respect to an index, as in the first example above), refer to
the

`Coordinate`

property.It can be useful to specify what is the implicit coordinate with respect to which a
derivative acts. For instance, for a derivative in the $\tau$ direction, we could
write

\tau::Coordinate;
\partial{#}::Derivative(\tau);
f::Depends(\tau);
ex:= \partial{f g};
unwrap(ex);

\(\displaystyle{}\text{Attached property Coordinate to }\tau.\)

\(\displaystyle{}\text{Attached property Derivative to }\partial{\#}.\)

\(\displaystyle{}\text{Attached property Depends to }f.\)

\(\displaystyle{}\partial\left(f g\right)\)

\(\displaystyle{}g \partial{f}\)