Use implicit differentiation to determine \( \frac{d y}{d x} \) for the equation \( y e^{5 x}+3 y-6=0 \) Solve the obtained equation \( e^{5 x} \frac{d y}{d x}+5 y e^{5 x}+3 \frac{d y}{d x}=0 \) for \( \frac{d y}{d x} \)

To use implicit differentiation on the equation \( y e^{5 x} + 3 y - 6 = 0 \), we first differentiate both sides with respect to \( x \). Recall that \( y \) is a function of \( x \) when we do this: \[ \frac{d}{dx}(y e^{5x}) + \frac{d}{dx}(3y) - \frac{d}{dx}(6) = 0. \] Using the product rule on \( ye^{5x} \) gives us: \[ \frac{d y}{d x} e^{5x} + y \cdot 5 e^{5x} + 3 \frac{d y}{d x} = 0. \] Now, we can rearrange the terms and factor out \( \frac{d y}{d x} \): \[ \frac{d y}{d x} e^{5x} + 3 \frac{d y}{d x} = -5 y e^{5x}. \] Combining \( \frac{d y}{d x} \): \[ \frac{d y}{d x} (e^{5x} + 3) = -5 y e^{5x}. \] Now, we can solve for \( \frac{d y}{d x} \): \[ \frac{d y}{d x} = \frac{-5y e^{5x}}{e^{5x} + 3}. \] And that is the expression for \( \frac{d y}{d x} \) in terms of \( y \) and \( x \)!