The derivative applies to constant functions, polynomials,
the exponential ^,the integral
powers ^&n,and those assigned by u`v D. n .
It also applies to functions derived from these by addition,
subtraction, multiplication, and division(u+v, etc.);
by the composition u@v;and by the
inverse u^:_1 .Since functions such as j.
and - (negation) and %: (square root) and 1&o.
(sin) and 6&o. (cosh) may all be so derived,
they are also in the domain of the derivative.
Others are treated by approximation.
The derivative of an arbitrary function may also be treated by
a polynomial approximation, (provided by the matrix divide),
or by approximations using the secant slope D: .
If the argument rank of u is a and the result
rank is r,then the argument rank of u D.1
is also a,but its result rank is r+a :
the result of u D.1 is the derivative of each atom of the
result of u with respect to each element of its argument,
giving what is commonly referred to as the partial derivatives.
For example:
The adverbs D=: 1 : 'u"0 D.1'
and VD=: 1 : 'u"1 D.1' assign ranks to their arguments,
then take the first derivative; they are convenient for use
in scalar and vector calculus: