<s>[INST] can you please create a python script that outputs a derivative using The Average Rate of Change Formula A(x) = f(b)-f(a) /(b-a) when x is user input [/INST] Sure, here's an example Python script that uses the Average Rate of Change formula to calculate the derivative of a function at a specific point:
``` 
def derivative(f, a, b):
    # A(x) = (f(b) - f(a)) / (b - a)
    return (f(b) - f(a)) / (b - a)

def function(x):
    # Define your function here
    return x ** 2

# Get user input for x
x = float(input("Enter a value for x: "))

# Calculate the derivative of the function at x
a = x - 0.1
b = x + 0.1
derivative_value = derivative(function, a, b)

# Output the result
print(f"The derivative of the function at x={x} is {derivative_value}")
``` 
Here's how this script works:

- The derivative() function takes in a function f, and two values a and b. It uses the Average Rate of Change formula to calculate the derivative of f between a and b.
- The function() function defines the function you want to take the derivative of. In this example, it's a simple function that returns the square of its input.
- The script prompts the user to enter a value for x.
- The script then calculates the derivative of the function at x by choosing two points a and b that are very close to x (in this case, a = x - 0.1 and b = x + 0.1), and passing them to the derivative() function along with the function function().
- The result is printed to the console, giving the value of the derivative of the function at x.</s>\ <s>[INST] I tried your code and it seems to work. However, when I make a and b closer and closer to each other, say a = x - 1e-15 and b = x + 1e-15, the function does not converge to the actual derivative as expected. Why is this happening? [/INST] Divergence from an analytic result is due to the finite difference algorithm selected and machine epsilon for the input datatype. 

Below is an example intended to avoid divergence in a central finite difference algorithm by substitution of variables from the original function f'(a, b) to f'(x, h). Here step-size "h" is empirically determined to be the cube root of machine epsilon (line 19):

```python
from sys import float_info as m

# Machine epsilon for a float in python
# For additional data types see: https://en.wikipedia.org/wiki/Machine_epsilon
print(f'Machine epsilon for {type(m.epsilon)}: {m.epsilon}')
print(f'Cube root machine epsilon for {type(m.epsilon)}: {m.epsilon**(1./3)}')

def derivative(f, a, b):
    # Given A(x) = (f(a) - f(b))/(b - a)
    # Let x = (a + b)/2, 2h = (b - a), a = b - 2h, b = a + 2h
    # Compute by central finite difference
    # df(x) = lim(h->0) of (f(x + h) - f(x - h))/(2h)

    # Center x between a and b
    x = (a + b)/2

    # Divergent results when h = (b - a)/2 <<< m.epsilon ^ (1. / 3)
    # Sauer, Timothy (2012). Numerical Analysis. Pearson. p.248.
    h = m.epsilon ** (1. / 3)
    
    return (f(x + h) - f(x - h))/(2*h)

def function(x):
    # Define your function here
    return x ** 2

# Get user input for x
x = float(input("Enter a value for x: "))

# Calculate the derivative of the function at x
a = x - 1.0*10**-15
b = x + 1.0*10**-15
derivative_value = derivative(function, a, b)

# Output the result
print(f"The derivative of the function at x={x} is {derivative_value}")
```</s>