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--- @class DerivativeExpression
--- An expression for a single-variable derivative of an expression.
--- @field symbol SymbolExpression
--- @field expression Expression
DerivativeExpression = {}
__DerivativeExpression = {}
----------------------------
-- Instance functionality --
----------------------------
-- Creates a new single-variable derivative operation with the given symbol and expression.
--- @param expression Expression
--- @param symbol Symbol
--- @return DerivativeExpression
function DerivativeExpression:new(expression, symbol)
local o = {}
local __o = Copy(__ExpressionOperations)
o.expression = Copy(expression)
o.symbol = symbol or SymbolExpression("x")
__o.__index = DerivativeExpression
__o.__tostring = function(a)
return '(d/d' .. tostring(a.symbol) .. " " .. tostring(a.expression) .. ')'
end
__o.__eq = function(a, b)
-- This shouldn't be needed, since __eq should only fire if both metamethods have the same function, but for some reason Lua always runs this anyway
if not b:type() == DerivativeExpression then
return false
end
return a.symbol == b.symbol and a.expression == b.expression
end
o = setmetatable(o, __o)
return o
end
--- @return Expression
function DerivativeExpression:evaluate()
local exp = self.expression
-- The derivative of a constant is 0
if exp:isconstant() then
return Integer.zero()
end
-- The derivative of a symbol is either 1 or 0
if exp:type() == SymbolExpression then
if self.symbol == exp then
return Integer.one()
end
return Integer.zero()
end
-- Chain rule for arbitrary functions
if exp:type() == FunctionExpression then
local results = {}
for index,expression in ipairs(exp.expressions) do
local dout = FunctionExpression(exp.name,exp.expressions,exp.derivatives)
dout.variables = exp.variables
dout.derivatives[index] = dout.derivatives[index]+Integer.one()
local dinn = DerivativeExpression(expression,self.symbol):evaluate()
results[index] = dout*dinn
end
return BinaryOperation(BinaryOperation.ADD,results):autosimplify()
end
--if exp:type() == FunctionExpression then
-- local results = {}
-- for index,expression in ipairs(exp.expressions) do
-- local inn = DerivativeExpression(expression,self.symbol):autosimplify()
-- local out = Copy(exp)
-- out.orders[index] = out.orders[index] + Integer.one()
-- local result = inn*out
-- table.insert(results,result)
-- end
-- return BinaryOperation(BinaryOperation.ADD,results):autosimplify()
--end
--if exp:type() == FunctionExpression then
-- if exp.expressions[2] then
-- return self
-- end
-- return DerivativeExpression(exp.expressions[1], self.symbol) * FunctionExpression(exp.name, exp.expressions, exp.orders[1] + Integer.one(), exp.variables[1]):autosimplify()
--end
-- Chain rule for trig functions
if exp:type() == TrigExpression then
local internal = DerivativeExpression(exp.expression, self.symbol)
if exp.name == "sin" then
return (internal * COS(exp.expression)):autosimplify()
end
if exp.name == "cos" then
return (internal * -SIN(exp.expression)):autosimplify()
end
if exp.name == "tan" then
return (internal * SEC(exp.expression)^Integer(2)):autosimplify()
end
if exp.name == "csc" then
return (internal * -CSC(exp.expression)*COT(exp.expression)):autosimplify()
end
if exp.name == "sec" then
return (internal * -SEC(exp.expression)*TAN(exp.expression)):autosimplify()
end
if exp.name == "cot" then
return (internal * -CSC(exp.expression)^Integer(2)):autosimplify()
end
if exp.name == "arcsin" then
return (internal / (Integer(1)-exp.expression^Integer(2))^(Integer(1)/Integer(2))):autosimplify()
end
if exp.name == "arccos" then
return (-internal / (Integer(1)-exp.expression^Integer(2))^(Integer(1)/Integer(2))):autosimplify()
end
if exp.name == "arctan" then
return (internal / (Integer(1)+exp.expression^Integer(2))):autosimplify()
end
if exp.name == "arccsc" then
return (-internal / (ABS(exp.expression) * (Integer(1)-exp.expression^Integer(2))^(Integer(1)/Integer(2)))):autosimplify()
end
if exp.name == "arcsec" then
return (internal / (ABS(exp.expression) * (Integer(1)-exp.expression^Integer(2))^(Integer(1)/Integer(2)))):autosimplify()
end
if exp.name == "arccot" then
return (-internal / (Integer(1)+exp.expression^Integer(2))):autosimplify()
end
end
-- TODO: Piecewise functions
if self:type() == AbsExpression then
return DerivativeExpression(self.expression, self.symbol):autosimplify()
end
-- Uses linearity of derivatives to evaluate sum expressions
if exp.operation == BinaryOperation.ADD then
local parts = {}
for i, expression in pairs(exp.expressions) do
parts[i] = DerivativeExpression(expression, self.symbol)
end
return BinaryOperation(BinaryOperation.ADD, parts):autosimplify()
end
-- Uses product rule to evaluate product expressions
if exp.operation == BinaryOperation.MUL then
local sums = {}
for i, expression in pairs(exp.expressions) do
local products = {}
for j, innerexpression in pairs(exp.expressions) do
if i ~= j then
products[j] = innerexpression
else
products[j] = DerivativeExpression(innerexpression, self.symbol)
end
end
sums[i] = BinaryOperation(BinaryOperation.MUL, products)
end
return BinaryOperation(BinaryOperation.ADD, sums):autosimplify()
end
-- Uses the generalized power rule to evaluate power expressions
if exp.operation == BinaryOperation.POW then
local base = exp.expressions[1]
local exponent = exp.expressions[2]
return BinaryOperation.MULEXP({
BinaryOperation.POWEXP({base, exponent}),
BinaryOperation.ADDEXP({
BinaryOperation.MULEXP({
DD(base, self.symbol),
BinaryOperation.DIVEXP({exponent, base})}),
BinaryOperation.MULEXP({
DD(exponent, self.symbol),
LN(base)})})
}):autosimplify()
end
if exp:type() == Logarithm then
local base = exp.base
local expression = exp.expression
return BinaryOperation.SUBEXP({
BinaryOperation.DIVEXP({DD(expression, self.symbol),
BinaryOperation.MULEXP({expression, LN(base)})}),
BinaryOperation.DIVEXP({
BinaryOperation.MULEXP({LN(expression), DD(base, self.symbol)}),
BinaryOperation.MULEXP({BinaryOperation.POWEXP({LN(base), Integer(2)}), base})
})
}):autosimplify()
end
return self
end
--- @return Expression
function DerivativeExpression:autosimplify()
return DerivativeExpression(self.expression:autosimplify(), self.symbol):evaluate()
end
--- @return table<number, Expression>
function DerivativeExpression:subexpressions()
return {self.expression}
end
--- @param subexpressions table<number, Expression>
--- @return DerivativeExpression
function DerivativeExpression:setsubexpressions(subexpressions)
return DerivativeExpression(subexpressions[1], self.symbol)
end
-- function DerivativeExpression:freeof(symbol)
-- return self.symbol.freeof(symbol) and self.expression:freeof(symbol)
-- end
-- Substitutes each expression for a new one.
-- function DerivativeExpression:substitute(map)
-- for expression, replacement in pairs(map) do
-- if self == expression then
-- return replacement
-- end
-- end
-- -- Typically, we only perform substitution on autosimplified expressions, so this won't get called. May give strange results, i.e.,
-- -- substituting and then evaluating the derivative may not return the same thing as evaluating the derivative and then substituting.
-- return DerivativeExpression(self.expression:substitute(map), self.symbol)
-- end
--- @param other Expression
--- @return boolean
function DerivativeExpression:order(other)
if other:type() == IntegralExpression then
return true
end
if other:type() ~= DerivativeExpression then
return false
end
if self.symbol ~= other.symbol then
return self.symbol:order(other.symbol)
end
return self.expression:order(other.expression)
end
--- @return string
function DerivativeExpression:tolatex()
return '\\frac{d}{d' .. self.symbol:tolatex() .. '}\\left(' .. self.expression:tolatex() .. '\\right)'
end
-----------------
-- Inheritance --
-----------------
__DerivativeExpression.__index = CompoundExpression
__DerivativeExpression.__call = DerivativeExpression.new
DerivativeExpression = setmetatable(DerivativeExpression, __DerivativeExpression)
----------------------
-- Static constants --
----------------------
DD = function(expression, symbol)
return DerivativeExpression(expression, symbol)
end
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