/*
* Differentiates one-variable functions with the substitution rule, which works as follows:
* First it computes the outer derivative.
* This is the derivative of the function with the argument substituted for a single variable.
* Then it multiplies this with the derivative of the substituted argument.
*/
public static string DifferentiateFunction(string functionName, string argument)
{
var outerDerivative = functionName switch
{
"sin" => StringMath.ToPrefixFormat(argument, "cos"),
"cos" => StringMath.MultiplyStrings("-1", StringMath.ToPrefixFormat(argument, "sin")),
"tan" => StringMath.DivideStrings("1", StringMath.PowerStrings(StringMath.ToPrefixFormat(argument, "cos"), "2")),
"ln" => StringMath.DivideStrings("1", argument),
"exp" => StringMath.ToPrefixFormat(argument, "exp"),
_ => throw new FormatException(),
};
return(StringMath.MultiplyStrings(Differentiate(argument), outerDerivative));
}
/*
* Differentiates an operator expression. Uses rules for operands being general diffentiable functions.
* +/-: Differentiation linear over +/-
* *: Chain rule
*./: https://www.wolframalpha.com/input/?i=differentiate+f%28x%29%2Fg%28x%29
* ^: https://www.wolframalpha.com/input/?i=differentiate+f%28x%29%5Eg%28x%29 (using expanded form)
*/
public static string DifferentiateOperator(string operatorSymbol, string arguments)
{
string[] operands = DifferentiationTokenizer.SplitArguments(arguments);
string firstDerivative = Differentiate(operands[0]);
string secondDerivative = Differentiate(operands[1]);
switch (operatorSymbol[0])
{
case '+':
case '-':
return(StringMath.AddStrings(firstDerivative, secondDerivative, operatorSymbol[0]));
case '*':
return(StringMath.AddStrings(StringMath.MultiplyStrings(firstDerivative, operands[1]), StringMath.MultiplyStrings(secondDerivative, operands[0])));
case '/':
return(StringMath.DivideStrings(
StringMath.AddStrings(
StringMath.MultiplyStrings(firstDerivative, operands[1]),
StringMath.MultiplyStrings(secondDerivative, operands[0]), '-'),
StringMath.PowerStrings(operands[1], "2")));
case '^':
return(StringMath.AddStrings(
StringMath.MultiplyStrings(
StringMath.MultiplyStrings(firstDerivative, operands[1]),
StringMath.PowerStrings(operands[0], StringMath.AddStrings(operands[1], "1", '-'))),
StringMath.MultiplyStrings(
StringMath.MultiplyStrings(secondDerivative, StringMath.LogarithmString(operands[0])),
StringMath.PowerStrings(operands[0], operands[1])
)
));
default:
throw new FormatException();
}
}
