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.

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which can be found using the derivative of the function.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is calculated as the limit of the average rate of change of the function over an interval as the interval approaches zero. For a function f(x), the derivative is denoted as f'(x) or df/dx.

Point-Slope Form

The point-slope form of a linear equation is used to write the equation of a line when the slope and a point on the line are known. It is expressed as y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful for finding the equation of a tangent line.