45–50. Tangent lines Carry out the following steps. <IMAGE>
b. Determine an equation of the line tangent to the curve at the given point.
x³+y³=2xy; (1, 1)

Verified step by step guidance

1. First, identify the given curve equation: \(x^3 + y^3 = 2xy\). We need to find the derivative to determine the slope of the tangent line at the point (1, 1).
2. Use implicit differentiation to differentiate both sides of the equation with respect to \(x\). Remember that \(y\) is a function of \(x\), so apply the chain rule when differentiating terms involving \(y\).
3. Differentiate the left side: \(\frac{d}{dx}(x^3) + \frac{d}{dx}(y^3) = 3x^2 + 3y^2 \frac{dy}{dx}\).
4. Differentiate the right side: \(\frac{d}{dx}(2xy) = 2y + 2x \frac{dy}{dx}\).
5. Set the derivatives equal: \(3x^2 + 3y^2 \frac{dy}{dx} = 2y + 2x \frac{dy}{dx}\). Solve for \(\frac{dy}{dx}\) to find the slope of the tangent line at the point (1, 1). Substitute \(x = 1\) and \(y = 1\) into the derivative to find the specific slope at that point. Finally, use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), to write the equation of the tangent line.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation, rather than explicitly as y = f(x). In this case, the equation x³ + y³ = 2xy involves both x and y, requiring us to differentiate both sides with respect to x while treating y as a function of x. This method allows us to find dy/dx, which is essential for determining the slope of the tangent line.

Tangent Line Equation

The equation of a tangent line at a given point on a curve can be expressed using the point-slope form: y - y₀ = m(x - x₀), where (x₀, y₀) is the point of tangency and m is the slope at that point. Once the derivative (slope) is calculated using implicit differentiation, this formula can be applied to find the specific equation of the tangent line at the point (1, 1) for the given curve.

Slope of the Tangent Line

The slope of the tangent line represents the instantaneous rate of change of the function at a specific point. In the context of the curve defined by the equation x³ + y³ = 2xy, the slope can be found by evaluating the derivative dy/dx at the point (1, 1). This slope is crucial for constructing the tangent line, as it indicates how steep the line will be at that point on the curve.