def poly_dev(poly, dev):
    """
    :param poly: a polynomial
    :param dev: derivative desired, e.g., 2nd
    """

    c = 0                                # correction
    r = dev                              # to remove
    while dev > 0:
        for i in range(1, len(poly)-c):
            poly[i-1] = (len(poly)-(i+c))*poly[i-1]
        dev -= 1                                    # I suspect this
        c += 1                                      # this can be simplified
    return poly[:-r]

>>> poly = [3, 5, 2, 2]
>>> print(poly_dev(poly, 2))
[18, 10]
>>> print(poly)
[18, 10, 2, 2]

def poly_dev(poly, dev):
    if dev == 0:
        return poly[:]
    if dev > len(poly):
        return list()
    if dev < 0:
        raise ValueError("negative derivative")
    p = 1
    for k in range(2, dev+1):
        p *= k
    poly = poly[:-dev]
    n = len(poly)-1
    for i in range(len(poly)):
        poly[n-i] *= p
        p = p * (i+dev+1) // (i+1)
    return poly

import functools
import operator

def product(start, end):
    return functools.reduce(operator.mul, range(start, end + 1), 1)

def poly_derivative(poly, n):
    if n < 0:
        raise ValueError("The nth order derivative must be non-negative")
    return [x * product(idx + 1, idx + n) for (idx, x) in enumerate(poly[n:])]

def poly_deriv(poly, nth=1):
    …

def poly_deriv(poly, nth=1):
    degree = len(poly) - 1
    if degree - nth < 0:
        return [0]   # or an empty list? Your decision.
    poly = poly[: -nth]
    for i in range(len(poly)):
        for exponent in range(degree - i, degree - i - nth, -1):
            poly[i] = exponent * poly[i]
    return poly