# Symbolic Differentiation

#### by gowtham 2010-02-09 09:59:45

Differentiate(input)
{
Stack = FillStack(input)
output = DifferentiateStack(Stack)
Optimize(output)
return output
}

FillStack(input)
{
operators[] { "+-", "*/", "^%" }

stack.push(input)
loop( n = 1 to stack.size() )
{
if stack[n] is not operator
if GetOperator(stack[n],
operators) success
{
Split stack[n]
stack.Insrt(left operand)
stack.Insrt(right operand)
}
else
GetFunction(stack[n])
}
}

DifferentiateStack(stack, index)
{
if stack[index] is operator
{
index++
u = stack[index]
du = DifferentiateStack(stack, index)
v = stack[index]
dv = DifferentiateStack(stack, index)

if operator = '-' or '+'
output = du+operator+dv
else if operator = '*'
output = u*dv+du*v
else if operator = '/'
output = (du*v-u*dv)/v^2
else if operator = '^'
output = v*u^(v-1)*du+u^v*ln(u)*dv
}
else
output = stack[index++].GetDifferentiation()

return output
}

void Optimize(str)
{
replace "--" with "" or "+"
replace "+-" with "-"
replace "((....))" with "(....)"
remove any 1*
remove any *1
remove any exponent equal 1
remove unneeded parentheses

if str changed then
Optimize(str)
}

ExpressionItem::GetDifferentiation()
{
if Function then
{
arument = Argument(input);
if function = SIN
output = Differentiate(arument)*cos(arument)
else if function = COS
output = Differentiate(arument)*(-sin(arument))
else
...
}
}
else
{
if input = "x"
output = "1"
else if input = "-x"
output = "-1"
else if input is numeric
output = "0"
else
output = "d"+input+"/dx"
}
}

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