We are given the equation $xe^y = \sin(y) + 4x$, and we need to find $\frac{dy}{dx}$. Let's solve this using implicit differentiation.

Step 1: Differentiate both sides of the equation with respect to $x$

Given equation: $xe^y = \sin(y) + 4x$

Using the product rule and chain rule:

• For the left-hand side $xe^y$, we apply the product rule: $\frac{d}{dx}(xe^y) = e^y \cdot \frac{d}{dx}(x) + x \cdot \frac{d}{dx}(e^y) = e^y + x e^y \frac{dy}{dx}$
• For the right-hand side $\sin(y) + 4x$, we use the chain rule: $\frac{d}{dx}(\sin(y)) = \cos(y) \frac{dy}{dx}, \quad \frac{d}{dx}(4x) = 4$

Thus, the differentiated equation is: $e^y + x e^y \frac{dy}{dx} = \cos(y) \frac{dy}{dx} + 4$

Step 2: Solve for $\frac{dy}{dx}$

First, collect all terms involving $\frac{dy}{dx}$ on one side of the equation: $x e^y \frac{dy}{dx} - \cos(y) \frac{dy}{dx} = 4 - e^y$

Factor out $\frac{dy}{dx}$: $\frac{dy}{dx} (x e^y - \cos(y)) = 4 - e^y$

Finally, solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{4 - e^y}{x e^y - \cos(y)}$

Step 3: Check the options

The correct option is:

$\boxed{\frac{dy}{dx} = \frac{e^y + x e^y - 4}{\cos(y)}}$

Would you like further details or explanations on any part? Here are some related questions:

1. How is implicit differentiation applied in other contexts?
2. What is the product rule, and when is it used?
3. How do you handle the chain rule in implicit differentiation?
4. What would change if $y$ was explicitly defined as a function of $x$?
5. Can you explain how to differentiate logarithmic or trigonometric functions using implicit differentiation?

Tip: When solving implicit differentiation problems, always ensure all terms involving $y$ are correctly handled using the chain rule!