Difference between defining a function inside versus outside of another function

Question

I have written a function derivative(w1, w2, pt) that evaluates the derivative of the function f(x) = w1 * x**3 + w2 * x - 1 at point pt. Strangely, I have found that I get different results depending on whether def f(x) is positioned inside or outside of derivative(w1, w2, pt). Why does the positioning of def f(x) matter / which is correct?

Example 1:

def derivative(w1, w2, pt):
    x = sy.Symbol('x')

    def f(x):
        return w1 * x**3 + w2 * x - 1

    # Get derivative of f(x)
    def df(x):
        return sy.diff(f(x),x)

    # Evaluate at point x
    return df(x).subs(x,pt)   

From which derivative(5, 8, 2) returns 68.

Example 2:

def f(x):
    return w1 * x**3 + w2 * x - 1

def derivative(w1, w2, pt):
    x = sy.Symbol('x')

    # Get derivative of f(x)
    def df(x):
        return sy.diff(f(x),x)

    # Evaluate at point x
    return df(x).subs(x,pt)

From which derivative(5, 8, 2) returns 53.

Answer 1

I think it's your global scope that is polluted. Look at this example :

import sympy as sy

def f(x, w1, w2):
    return w1 * x**3 + w2 * x - 1

def derivative(w1, w2, pt):
    x = sy.Symbol('x')

    # Get derivative of f(x)
    def df(x, w1, w2):
        return sy.diff(f(x, w1, w2),x)

    # Evaluate at point x
    return df(x, w1, w2).subs(x,pt)

print(derivative(5, 8, 2))

This is just a modified version of your example 2 and it returns the same answer.