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5.1.22 diff

Syntax:
diff ( poly_expression, ring_variable )
diff ( vector_expression, ring_variable )
diff ( ideal_expression, ring_variable )
diff ( module_expression, ring_variable )
diff ( matrix_expression, ring_variable )
Type:
the same as the type of the first argument
Syntax:
diff ( ideal_expression, ideal_expression )
Type:
matrix
Syntax:
diff ( number_expression, ring_parameter )
Type:
number
Purpose:
computes the partial derivative of a polynomial object by a ring variable (first forms)
respectively differentiates each polynomial (1..n) of the second ideal by the differential operator corresponding to each polynomial (1..m) in the first ideal, producing an m x n matrix.
respectively if the coefficient ring is a transcendental field extension, differentiates a number (that is, a rational function) by a transcendental variable (ring parameter).
Example:
 
  ring r=0,(x,y,z),dp;
  poly f=2x3y+3z5;
  diff(f,x);
==> 6x2y
  vector v=[f,y2+z];
  diff(v,z);
==> 15z4*gen(1)+gen(2)
  ideal j=x2-yz,xyz;
  ideal i=x2,x2+yz,xyz;
  // corresponds to differential operators
  // d2/dx2, d2/dx2+d2/dydz, d3/dxdydz:
  print(diff(i,j));
==> 2,0,
==> 1,x,
==> 0,1 
  // differentiation of rational functions:
  ring R=(0,t),(x),dp;
  number f = t^2/(1-t)^2;
  diff(f,t);
==> (-2t)/(t3-3t2+3t-1)
See contract; ideal; jacob; matrix; module; poly; var; vector.

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