// This file contains transformation rules suitable for finding // derivatives of expressions. transformations derivatives { // This is a simple program that lets you define transformation rules // and mathematical simplification rules and test them easily. // Derivative of both sides of an equation. D[_a === _b, _x] <-> D[_a, _x] === D[_b, _x] // Derivative as inverse of integration. D[Integrate[_f, _x], _x] <-> _f // Multiple derivatives // Bail-out condition for first derivative D[_n, _x, 1] <-> D[_n, _x] // Bail-out condition for zeroth derivative D[_n, _x, 0] <-> _n // Otherwise take one derivative and decrement. D[_n, _x, _times is isInteger] <-> D[D[_n,_x], _x, _times-1] // Degenerate cases D[_c, _x] :: freeOf[_c, _x] <-> 0 D[_x, _x] <-> 1 // The following are shortcuts and aren't strictly needed, but they're // closer to what a human would do and make the transformation path simpler. // The constraints are necessary to prevent naive evaluation of, say, x^x. D[(_c:1) _x^(_y:1), _x] :: freeOf[_c, _x] && freeOf[_y, _x] <-> (_c _y) _x^(_y-1) //D[_a^_x, _x] <-> _a^_x ln[_a] D[sin[_x], _x] <-> cos[_x] D[cos[_x], _x] <-> -sin[_x] D[tan[_x], _x] <-> 1/cos[_x]^2 D[arcsin[_x], _x] <-> 1/sqrt[1 - _x^2] D[arccos[_x], _x] <-> -1/sqrt[1 - _x^2] D[arctan[_x], _x] <-> 1/(1 + _x^2) D[arccsc[_x], _x] <-> -1/(magnitude[_x] / sqrt[_x^2 - 1]) D[arcsec[_x], _x] <-> 1/(magnitude[_x] / sqrt[_x^2 - 1]) D[arccot[_x], _x] <-> -1/(1 + _x^2) D[sinh[_x], _x] <-> cosh[_x] D[cosh[_x], _x] <-> sinh[_x] D[tanh[_x], _x] <-> 1/cosh[_x]^2 D[arcsinh[_x], _x] <-> 1/sqrt[_x^2 + 1] D[arccosh[_x], _x] <-> -1/sqrt[_x^2 - 1] D[arctanh[_x], _x] <-> 1/(1 - _x^2) D[ln[_x], _x] <-> 1/_x D[e^_x, _x] <-> e^_x // Chained derivative rules D[_a + _b, _x] <-> D[_a,_x] + D[_b,_x] // These rules can loop if _u or _v equals _x. // Prevent that? Need excluding match? D[_u _v, _x] <-> _u D[_v, _x] + _v D[_u, _x] D[_u^_v, _x] <-> _v _u^(_v-1) D[_u, _x] + _u^_v ln[_u] D[_v,_x] // This matches a function that depends on x. The excluding structureEquals // test keeps it from looping forever when _u equals _x D[_f[_u], _x] :: expressionContains[_u, _x] && ! structureEquals[_u, _x] <-> D[_u, _x] D[_f[_u], _u] // This matches a function that does not depend on x D[_f[_u], _x] :: freeOf[_u, _x] <-> 0 }