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prolog rules for differential calculus symbolic computation
/****************** Differential Calculus ***************/
d( X, X, 1 ):- !. /* d(X) w.r.t. X is 1 */
d( C, X, 0 ):- atomic(C). /* If C is a constant then */
/* d(C)/dX is 0 */
d( U+V, X, R ):- /* d(U+V)/dX = A+B where */
d( U, X, A ), /* A = d(U)/dX and */
d( V, X, B ),
R = A + B.
d( U-V, X, R ):-
d( U, X, A ),
d( V, X, B ),
R = A - B.
d( C*U, X, R ):-
atomic(C),
C \= X,
d( U, X, A ),
R = C * A,
!.
d( U*V, X, U*B+V*A ):- /* d(U*V)/dX = B*U+A*V where */
d( U, X, A ), /* A = d(U)/dX and */
d( V, X, B ). /* B = d(V)/dX */
d( U/V, X, (A*V-B*U)/(V^2) ):- /* d(U/V)/dX = (A*V-B*U)/(V*V) */
d( U, X, A), /* where A = d(U)/dX and */
d( V, X, B). /* B = d(V)/dX */
d( U^C, X, R ):- /* d(U^C)/dX = C*A*U^(C-1) */
atomic(C), /* where C is a number or */
C\=X,
d( U, X, A ),
R = C * A * U ^ ( C - 1 ).
d( sin(W), X, Z*cos(W) ):- /* d(sin(W))/dX = Z*cos(W) */
d( W, X, Z). /* where Z = d(W)/dX */
d( exp(W), X, Z*exp(W) ):- /* d(exp(W))/dX = Z*exp(W) */
d( W, X, Z). /* where Z = d(W)/dX */
d( log(W), X, Z/W ):- /* d(log(W))/dX = Z/W */
d( W, X, Z). /* where Z = d(W)/dX */
d( cos(W), X, -(Z*sin(W)) ):- /* d(cos(W))/dX = Z*sin(W) */
d( W, X, Z). /* where Z = d(W)/dX */
d( tan(W), X, (Z*sec(W)^2) ):- /* d(tan(W))/dX = Z*sec(W)^2 */
d( W, X, Z). /* where Z = d(W)/dX */
d( cot(W), X, -(Z*cosec(W)^2) ):- /* d(cot(W))/dX = -Z*cosec(W)^2 */
d( W, X, Z). /* where Z = d(W)/dX */
d( sec(W), X, (Z*sec(W)*tan(W)) ):- /* d(sec(W))/dX = sec(W)*tan(W) */
d( W, X, Z). /* where Z = d(W)/dX */
d( cosec(W), X, -(Z*cosec(W)*cot(W)) ):- /* d(cosec(W))/dX = -cosec(W)*cot(W) */
d( W, X, Z). /* where Z = d(W)/dX */
d( arcsin(W), X, Z/sqrt(1-W^2) ):- /* d(arcsin(W))/dX = Z/sqrt(1-W^2) */
d( W, X, Z). /* where Z = d(W)/dX */
d( arccos(W), X, -(Z/sqrt(1-W^2)) ):- /* d(arccos(W))/dX = -(Z/sqrt(1-W^2) )*/
d( W, X, Z). /* where Z = d(W)/dX */
d( arctan(W), X, Z/(1+W^2) ):- /* d(arctan(W))/dX = Z/(1+W^2) */
d( W, X, Z). /* where Z = d(W)/dX */
d( arccot(W), X, -(Z/(1+W^2)) ):- /* d(arccot(W))/dX = -(Z/(1+W^2)) */
d( W, X, Z). /* where Z = d(W)/dX */
d( arcsec(W), X, (Z/(W*sqrt(W^2-1))) ):- /* d(arcsec(W))/dX = (Z/(W*sqrt(W^2-1))) */
d( W, X, Z). /* where Z = d(W)/dX */
d( arccosec(W), X, -(Z/(W*sqrt(W^2-1))) ):- /* d(arccosec(W))/dX = -(Z/(W*sqrt(W^2-1))) */
d( W, X, Z). /* where Z = d(W)/dX */
/****************** End Differential Calculus ***************/
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