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// 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
}
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This is a program written in the programming language Frink.
For more information, view the Frink
Documentation or see More Sample Frink Programs.
Alan Eliasen was born 19966 days, 1 hours, 29 minutes ago.