Use implicit differentiation to find dy/dx.
x³ = (x + y) / (x - y)

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. The equation is x^3 = (x + y) / (x - y).
2. Apply the power rule to differentiate x^3 with respect to x, which gives 3x^2.
3. For the right side, use the quotient rule for differentiation. If u = x + y and v = x - y, then the derivative of u/v is (v * du/dx - u * dv/dx) / v^2.
4. Differentiate u = x + y with respect to x, which gives du/dx = 1 + dy/dx. Differentiate v = x - y with respect to x, which gives dv/dx = 1 - dy/dx.
5. Substitute du/dx and dv/dx into the quotient rule formula and set the derivatives equal: 3x^2 = ((x - y)(1 + dy/dx) - (x + y)(1 - dy/dx)) / (x - y)^2. Solve for dy/dx.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for y in terms of x, we differentiate both sides of the equation with respect to x, treating y as a function of x. This allows us to find dy/dx without isolating y, which is particularly useful for complex relationships.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When using implicit differentiation, we apply the chain rule to account for the derivative of y with respect to x, denoted as dy/dx. This means that when differentiating terms involving y, we multiply by dy/dx to reflect the dependency of y on x.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate variables or terms. In the context of implicit differentiation, after differentiating both sides of the equation, we often need to manipulate the resulting expression to solve for dy/dx. This may include combining like terms, factoring, or moving terms across the equation to achieve the desired form.