create a function that performs newton's method without the user inputting the derivative in javascript

Newton's method is used to find the root of a function by using iterative steps. The derivative of the function is required for this method. However, if the derivative is not provided, the derivative approximation can be used as a substitute.

Here's an example of how you can create a function that performs Newton's method without the user inputting the derivative in JavaScript:

index.tsx
function newtonsMethodWithoutDerivative(func, initialGuess, epsilon) {
  const h = 0.000001; // small value for calculating the derivative approximation

  let x = initialGuess;
  let prevX = x - 1; // initialize the previous value to a different value than x
  while (Math.abs(x - prevX) > epsilon) {
    prevX = x;
    const f = func(prevX);
    const fPrime = (func(prevX + h) - func(prevX)) / h; // derivative approximation
    x = prevX - f / fPrime;
  }

  return x;
}
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In this code, the newtonsMethodWithoutDerivative function takes three parameters: func (the function for which we want to find the root), initialGuess (the initial guess for the root), and epsilon (the desired degree of accuracy).

The function then performs iterations using Newton's method until the difference between consecutive approximations is smaller than epsilon. It calculates the derivative approximation using a small value h and updates the approximation for the root x using the formula: x = prevX - f / fPrime, where f is the function value and fPrime is the derivative approximation.

This implementation makes an assumption that the provided function is differentiable and that the derivative approximation is a good enough substitute. However, it is important to note that Newton's method may not converge or may converge to a different root if the derivative approximation is not accurate.

Remember to test the function thoroughly before using it in a production environment.