The following equations implicitly define one or more functions.
a. Find dy/dx using implicit differentiation.
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

Verified step by step guidance

1. Start by rewriting the given equation as suggested: \( y^3 - 1 = xy - x \). This will help in factoring both sides.
2. Factor both sides of the equation. The left side \( y^3 - 1 \) can be factored as \( (y - 1)(y^2 + y + 1) \). The right side \( xy - x \) can be factored as \( x(y - 1) \).
3. Now, set the factored forms equal to each other: \( (y - 1)(y^2 + y + 1) = x(y - 1) \). Notice that \( y - 1 \) is a common factor on both sides.
4. Divide both sides by \( y - 1 \) (assuming \( y \neq 1 \) to avoid division by zero), which simplifies the equation to \( y^2 + y + 1 = x \).
5. Differentiate both sides with respect to \( x \) using implicit differentiation. For the left side, differentiate \( y^2 + y + 1 \) with respect to \( x \) using the chain rule, and for the right side, differentiate \( x \) directly. This will give you an expression for \( \frac{dy}{dx} \).

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, applying the chain rule when necessary. This method is particularly useful for equations that define y implicitly, allowing us to find dy/dx without isolating y.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential when applying implicit differentiation, as it helps manage the derivatives of terms involving y.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that can be multiplied together to obtain the original expression. In the context of the given problem, factoring can simplify the equation after rewriting it, making it easier to differentiate. Understanding how to factor polynomials and other expressions is crucial for manipulating equations effectively in calculus.