// 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]

   // 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[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]
   D[_f[_u], _x]     <-> D[_u, _x] D[_f[_u], _u]
}