var
  f, df: string;
  t: TTranslator;
begin
  t:= TTranslator.Create;
  f:= 'x^2*e^(x/3)';
  df:= t.Derivative(f, 'x');
  //...
end;

f(x)       : x^2*e^(x/3)
df/dx (old): (2*x)*e^(x/3)+(e^(x/3)*(1/3))*x^2
df/dx (2.3): 2*x*e^(x/3)+e^(x/3)/3*x^2

f(x)       : 2*x^3+A/4*x^2-x/3
df/dx (old): (3*x^2)*2+((2*x)*A)/4-1/3
df/dx (2.3): 6*x^2+1/2*x*A-1/3

f(x)       : (A/2)*arctan(x^2)
df/dx (old): ((1/(1+(x^2)^2))*(2*x))*(A/2)
df/dx (2.3): x*A/(1+x^4)

f(x)       : P{n-1 m+2}(x)
df/dx (old): ((((n-1)+1)-(m+2))*P{(n-1)+1 m+2}(x) -(((n-1)+1)*x)*P{n-1 m+2}(x))/(x^2-1)
df/dx (2.3): ((n-m-2)*P{n m+2}(x)-n*x*P{n-1 m+2}(x))/(x^2-1)