11. Explain how to use implicit differentiation to find \( \frac{d y}{d x} \) for each of the following equations. Then find \( \frac{d y}{d x} \). (a) \( \quad-5 x^{4}-2 e^{y}=-5 y^{3}-7 \)

To use implicit differentiation, start by differentiating both sides of the equation with respect to \( x \). When you differentiate terms involving \( y \), remember to multiply by \( \frac{d y}{d x} \) because \( y \) is a function of \( x \). For the equation \( -5 x^4 - 2 e^y = -5 y^3 - 7 \), differentiate each term: the left-hand side becomes \( -20 x^3 - 2 e^y \frac{d y}{d x} \), and the right-hand side becomes \( -15 y^2 \frac{d y}{d x} \). Setting both differentiated sides equal gives you \( -20 x^3 - 2 e^y \frac{d y}{d x} = -15 y^2 \frac{d y}{d x} \). Now, collect all \( \frac{d y}{d x} \) terms on one side: \( -2 e^y \frac{d y}{d x} + 15 y^2 \frac{d y}{d x} = 20 x^3 \). Factor out \( \frac{d y}{d x} \) to solve for it: \( \frac{d y}{d x} = \frac{20 x^3}{15y^2 + 2 e^y} \). Voilà! You have found \( \frac{d y}{d x} \)! For some extra fun, did you know that implicit differentiation is used in various fields like physics and economics? Whether you're analyzing the trajectory of an object or determining the relationship between quantities in supply and demand curves, it's an essential tool that can simplify complex relationships without explicitly solving for \( y \). Additionally, a common mistake in implicit differentiation is neglecting to apply the chain rule correctly. When differentiating terms, particularly those involving \( y \), be sure to remember to multiply by \( \frac{d y}{d x} \) to capture the underlying relationship between \( x \) and \( y \). Forgetting this step can lead to incorrect derivatives!