69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.
x² = -6y; (-6, -6)

Verified step by step guidance

1. Rewrite the given curve equation \(x^{2} = -6y\) to express \(y\) explicitly in terms of \(x\). This will help in differentiating \(y\) with respect to \(x\).
2. Differentiate both sides of the equation \(x^{2} = -6y\) implicitly with respect to \(x\). Remember to apply the chain rule when differentiating \(y\) since \(y\) is a function of \(x\).
3. After differentiating, solve for \(\frac{dy}{dx}\), which represents the slope of the tangent line at any point on the curve.
4. Substitute the given point \((-6, -6)\) into the expression for \(\frac{dy}{dx}\) to find the slope of the tangent line at that specific point.
5. Use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope found in the previous step and \((x_1, y_1)\) is the given point, to write the equation of the tangent line.

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.

Slope of the Tangent Line

The slope of the tangent line at a given point on a curve is the value of the derivative dy/dx evaluated at that point. It represents the instantaneous rate of change of y with respect to x and determines the steepness and direction of the tangent line.

Equation of a Tangent Line

Once the slope of the tangent line and the point of tangency are known, the equation of the tangent line can be found using the point-slope form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point on the curve.