Derivative of Polynomial Product Using Product Rule (Medium) — Practice with Code Visualizer

In calculus and machine learning, computing derivatives is a foundational operation. The product rule is a fundamental differentiation technique that allows us to find the derivative of a product of two functions. Given two differentiable functions f(x) and g(x), the derivative of their product is:

$$(f \cdot g)' = f' \cdot g + f \cdot g'$$

Polynomial Representation: In this problem, polynomials are represented as coefficient lists where the index corresponds to the power of x. For example:

[3, 2, 1] represents the polynomial 3 + 2x + x²
[1, 0, 5] represents the polynomial 1 + 0x + 5x² = 1 + 5x²
[7] represents the constant polynomial 7

Mathematical Background: To differentiate a polynomial represented as coefficients, each term (a_i x^i) becomes (i \cdot a_i x^{i-1}). The product of two polynomials can be computed using convolution of their coefficient vectors.

Your Task: Implement a function that takes the coefficient representations of two polynomials f(x) and g(x), and returns the coefficient list representing the derivative of their product (f · g)'.

Output Requirements:

All coefficients in the result must be rounded to 4 decimal places
Trailing zero coefficients should be removed from the result
If the entire derivative is the zero polynomial, return [0.0]

Key Insight: While you could multiply the polynomials first and then differentiate, using the product rule directly (computing f' · g + f · g') is often more numerically stable and conceptually aligns with how automatic differentiation systems operate in deep learning frameworks.

Polynomials: • f(x) = 1 + 2x • g(x) = 3 + 4x

Derivatives: • f'(x) = 2 • g'(x) = 4

Using the Product Rule: (f · g)' = f' · g + f · g'

Step 1: f' · g = 2 · (3 + 4x) = 6 + 8x → coefficients [6, 8] Step 2: f · g' = (1 + 2x) · 4 = 4 + 8x → coefficients [4, 8] Step 3: Sum the results: [6+4, 8+8] = [10, 16]

Verification: The product f·g = (1+2x)(3+4x) = 3 + 4x + 6x + 8x² = 3 + 10x + 8x². The derivative is 10 + 16x, confirming our answer of [10.0, 16.0].

Polynomials: • f(x) = 1 + x + x² • g(x) = 2 + 3x

Derivatives: • f'(x) = 1 + 2x → coefficients [1, 2] • g'(x) = 3 → coefficients [3]

Using the Product Rule: Step 1: f' · g = (1 + 2x)(2 + 3x) = 2 + 3x + 4x + 6x² = 2 + 7x + 6x² → [2, 7, 6] Step 2: f · g' = (1 + x + x²)(3) = 3 + 3x + 3x² → [3, 3, 3] Step 3: Sum: [2+3, 7+3, 6+3] = [5, 10, 9]

The resulting derivative polynomial is 5 + 10x + 9x².

Polynomials: • f(x) = 5 (constant polynomial) • g(x) = 1 + 2x

Derivatives: • f'(x) = 0 • g'(x) = 2

Using the Product Rule: Step 1: f' · g = 0 · (1 + 2x) = 0 → [0] Step 2: f · g' = 5 · 2 = 10 → [10] Step 3: Sum: [0 + 10] = [10]

The derivative is the constant 10, represented as [10.0].