deriv

Function: deriv
Section: polynomials
C-Name: deriv
Prototype: GDn
Help: deriv(x,{v}): derivative of x with respect to v, or to the main
 variable of x if v is omitted.
Doc: derivative of $x$ with respect to the main
 variable if $v$ is omitted, and with respect to $v$ otherwise. The derivative
 of a scalar type is zero, and the derivative of a vector or matrix is done
 componentwise. One can use $x'$ as a shortcut if the derivative is with
 respect to the main variable of $x$; and also use $x''$, etc., for multiple
 derivatives altough \kbd{derivn} is often preferrable.

 By definition, the main variable of a \typ{POLMOD} is the main variable among
 the coefficients from its two polynomial components (representative and
 modulus); in other words, assuming a polmod represents an element of
 $R[X]/(T(X))$, the variable $X$ is a mute variable and the derivative is
 taken with respect to the main variable used in the base ring $R$.

 \bprog
 ? f = (x/y)^5;
 ? deriv(f)
 %2 = 5/y^5*x^4
 ? f'
 %3 = 5/y^5*x^4
 ? deriv(f, 'x) \\ same since 'x is the main variable
 %4 = 5/y^5*x^4
 ? deriv(f, 'y)
 %5 = -5/y^6*x^5
 @eprog

 This function also operates on closures, in which case the variable
 must be omitted. It returns a closure performing a numerical
 differentiation as per \kbd{derivnum}:
 \bprog
 ? f(x) = x^2;
 ? g = deriv(f)
 ? g(1)
 %3 = 2.0000000000000000000000000000000000000
 ? f(x) = sin(exp(x));
 ? deriv(f)(0)
 %5 = 0.54030230586813971740093660744297660373
 ? cos(1)
 %6 = 0.54030230586813971740093660744297660373
 @eprog