Derivative of Polynomial Ratio Using Division Rule

In calculus, the quotient rule provides a systematic method for computing the derivative of a function that is expressed as the ratio of two differentiable functions. This rule is essential when dealing with rational functions—quotients of polynomials—and has widespread applications in optimization, physics, and machine learning.

Given two polynomial functions g(x) (numerator) and h(x) (denominator), the quotient rule states that the derivative of their ratio f(x) = g(x) / h(x) is computed as:

$$f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{[h(x)]^2}$$

This formula elegantly combines the derivatives of both the numerator and denominator polynomials, weighted by the cross-product of the original functions.

Polynomial Representation: Polynomials are represented by their coefficients in descending order of powers. For example:

• [1, 2, 3] represents the polynomial x² + 2x + 3
• [2, -1, 0, 5] represents 2x³ - x² + 5
• [1, 0] represents x

Polynomial Evaluation and Differentiation: To evaluate a polynomial at a point x, substitute the value into the expression. To differentiate a polynomial:

• The derivative of aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ is n·aₙxⁿ⁻¹ + (n-1)·aₙ₋₁xⁿ⁻² + ... + a₁
• Each term's power decreases by 1, and the coefficient is multiplied by the original power

Your Task: Implement a function that computes the derivative of the quotient f(x) = g(x) / h(x) at a specified point x using the quotient rule. The function should:

1. Accept the coefficients of the numerator polynomial g(x)
2. Accept the coefficients of the denominator polynomial h(x)
3. Accept a point x at which to evaluate the derivative
4. Return the computed value of f'(x) at the given point

Assumption: The denominator polynomial h(x) is guaranteed to be non-zero at the evaluation point x.