Is there any better way to find nth derivative of a function in C?

Question

I want to write a subroutine that calculates the nth derivative of a function given by the subroutine of the form:

double func(double x)
{
    // a mathematical expression representing a smooth curve 
    //returns the value of the function at x      
}

i wrote the following subroutine:

double nthderive(double (*f)(double),double x0,int n)
{   
   const double delta=1.0e-8;
   double x1=x0 - delta;
   double x2=x0 + delta;
   if(n==0) {
       return f(x0);
   }
   else {
       return (nthderive(f,x2,n-1)-nthderive(f,x1,n-1))/(2*delta);
   }
}

Can someone suggest a better algorithm for finding the nth derivative?

Answer 1

The nth derivative can be computed using Cauchy's integral lemma.

Transform the integral over the path into a "standard integral" (see Line integral).

Then you can integrate that expression (e.g. with trapezoidal rule).

If you want to calculate high order derivatives this method will most likely be stable.