Tangent lines Determine an equation of the line tangent to the graph of y=(x²−1)² / x³−6x−1 at the point (0,−1).

Verified step by step guidance

1. Step 1: Identify the function and the point of tangency. The function is \( y = \frac{(x^2 - 1)^2}{x^3 - 6x - 1} \) and the point of tangency is \((0, -1)\).
2. Step 2: Find the derivative of the function \( y \) with respect to \( x \) to determine the slope of the tangent line. Use the quotient rule: \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \), where \( u = (x^2 - 1)^2 \) and \( v = x^3 - 6x - 1 \).
3. Step 3: Calculate \( u' \) and \( v' \). For \( u = (x^2 - 1)^2 \), use the chain rule: \( u' = 2(x^2 - 1) \cdot 2x = 4x(x^2 - 1) \). For \( v = x^3 - 6x - 1 \), \( v' = 3x^2 - 6 \).
4. Step 4: Substitute \( u, u', v, \) and \( v' \) into the quotient rule formula to find \( y' \). Simplify the expression to find the derivative.
5. Step 5: Evaluate the derivative \( y' \) at \( x = 0 \) to find the slope of the tangent line at the point \((0, -1)\). Use the point-slope form of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the point of tangency, to write the equation of the tangent line.