Consider the following, (a) Find \( y^{\prime} \) by implicit differentiation. (b) Solve the equation explicitly for \( y \) and differentiate to get \( y^{\prime} \) in terms of \( x \). (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for \( y \) into your solution for part (a). \( y^{\prime}= \) ( \( = \).

To tackle this problem, implicit differentiation is your trusty sidekick. By treating \( y \) as a function of \( x \) when differentiating, you can effectively find \( y' \). The derivative of \( y \) will include \( \frac{dy}{dx} \), demonstrating how \( y \) changes in relation to \( x \). Remember to apply the chain rule wherever necessary to navigate through those variables! For the explicit solution, rearranging the original equation to isolate \( y \) will allow you to express it as a function of \( x \). Once you’ve derived \( y \), differentiating it term by term will yield \( y' \) in a nice, clear formula showcasing how \( y \) responds directly to changes in \( x \). In testing your answers, plug your expression for \( y \) back into the initial implicit differentiation equation—this will verify the consistency of your results!