Derivatives Find and simplify the derivative of the following functions.
h(x) = (x−1)(2x²-1) / (x³-1)

Verified step by step guidance

1. Step 1: Identify the function as a quotient of two functions, where the numerator is \((x-1)(2x^2-1)\) and the denominator is \(x^3-1\). We will use the quotient rule for derivatives, which is given by \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}\), where \(u\) is the numerator and \(v\) is the denominator.
2. Step 2: Differentiate the numerator \(u = (x-1)(2x^2-1)\) using the product rule. The product rule states that \(\frac{d}{dx}(fg) = f'g + fg'\). Let \(f(x) = x-1\) and \(g(x) = 2x^2-1\). Find \(f'(x)\) and \(g'(x)\), then apply the product rule.
3. Step 3: Differentiate the denominator \(v = x^3-1\). The derivative of \(v\) is \(v' = 3x^2\).
4. Step 4: Substitute \(u\), \(u'\), \(v\), and \(v'\) into the quotient rule formula. Simplify the expression obtained from the quotient rule.
5. Step 5: Simplify the resulting expression further, if possible, by combining like terms and reducing any common factors in the numerator and the denominator.