69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.
y² - x²/64 = 1; (6, -5/4)

Verified step by step guidance

1. Recognize that the given curve is an implicit function defined by the equation \(y^{2} - \frac{x^{2}}{64} = 1\). Since \(y\) is not explicitly solved for \(x\), we will use implicit differentiation to find \(\frac{dy}{dx}\), the slope of the tangent line.
2. Differentiate both sides of the equation with respect to \(x\). Remember to apply the chain rule when differentiating \(y^{2}\), treating \(y\) as a function of \(x\): \(\frac{d}{dx}\left(y^{2}\right) - \frac{d}{dx}\left(\frac{x^{2}}{64}\right) = \frac{d}{dx}(1)\) which becomes \$2y \cdot \frac{dy}{dx} - \frac{2x}{64} = 0$.
3. Solve the resulting equation for \(\frac{dy}{dx}\) to express the slope of the tangent line in terms of \(x\) and \(y\): \$2y \cdot \frac{dy}{dx} = \frac{2x}{64}$ so \(\frac{dy}{dx} = \frac{2x}{64 \cdot 2y} = \frac{x}{64y}\).
4. Substitute the given point \((6, -\frac{5}{4})\) into the expression for \(\frac{dy}{dx}\) to find the slope of the tangent line at that point: \(m = \frac{6}{64 \cdot (-\frac{5}{4})}\).
5. Use the point-slope form of the equation of a line to write the equation of the tangent line: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (6, -\frac{5}{4})\) and \(m\) is the slope found in the previous step.

Key Concepts

Implicit Differentiation

Implicit differentiation is used when a function is given in an implicit form, such as an equation involving both x and y. Instead of solving for y explicitly, we differentiate both sides with respect to x, treating y as a function of x, and apply the chain rule to find dy/dx.

Equation of a Tangent Line

The equation of a tangent line at a point on a curve is given by y - y₁ = m(x - x₁), where m is the slope of the tangent line at that point. Once the derivative dy/dx is found, substituting the point's coordinates gives the slope needed to write the tangent line equation.

Hyperbola and Its Properties

The given curve y² - x²/64 = 1 represents a hyperbola. Understanding its shape and properties helps in visualizing the curve and verifying the correctness of the tangent line. The hyperbola's implicit form requires implicit differentiation to find slopes at specific points.