Intro

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The Derivative as a Limit

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A derivative is a rate: how fast the output moves when you nudge the input.

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

You cannot take a limit on a computer, so you pick a small $h$ and evaluate the fraction. But there are two ways to do it, and the choice matters more than it looks:

forward, error O (h) \frac{f ( x + h ) - f ( x )}{h} central, error O (h^{2}) \frac{f ( x + h ) - f ( x - h )}{2 h}

The central difference looks the same distance either side and is dramatically more accurate — with $h = 1 0^{- 5}$ the forward difference is right to about 5 digits, the central one to about 10. It is what every gradient checker uses.

f(x) = x^2,  f'(x) = 2x,  at x = 3 the answer is 6

forward, h=1e-5: ((3.00001)^2 - 3^2)/1e-5     = 6.00001
central, h=1e-5: ((3.00001)^2 - (2.99999)^2)/2e-5 = 6.00000

Do not make $h$ tiny to compensate. Below about $1 0^{- 8}$ the two function values agree to within floating-point noise, you subtract two nearly equal numbers, and the accuracy collapses. Around $1 0^{- 5}$ is the sweet spot.

Your task:

Implement central_difference(coeffs, x, h), where coeffs describes a polynomial and the derivative is estimated at x.

coeffs[i] is the coefficient of $x^{i}$ , so [1, 0, 2] means $1 + 2 x^{2}$ . Evaluate with np.polyval(coeffs[::-1], x), which expects the highest power first.

Example Tests

x squared has slope 2x, which is 6 at x = 3

Input: {"h":0.00001,"x":3,"coeffs":[0,0,1]}

Expected: 6

A straight line has the same slope everywhere

Input: {"h":0.00001,"x":100,"coeffs":[4,2]}

Expected: 2

A constant does not change, so its derivative is zero

Input: {"h":0.00001,"x":2,"coeffs":[7]}

Expected: 0

Python

import numpy as np


def central_difference(coeffs, x, h):
    """
    Numerical derivative of a polynomial by central difference.

    Args:
        coeffs: coefficients low power first, so [1, 0, 2] is 1 + 2x^2
        x:      the point to differentiate at
        h:      the step size

    Returns:
        float: (f(x+h) - f(x-h)) / (2h)
    """
    # YOUR CODE HERE
    pass

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