We are given the equation:

$- e^y \sin(x) = xy - 5y$

Our goal is to find $\frac{dy}{dx}$. Since $y$ is a function of $x$, we will differentiate both sides of the equation with respect to $x$, applying implicit differentiation.

Step 1: Differentiate the left-hand side

$\frac{d}{dx}\left( - e^y \sin(x) \right)$

This is a product of two functions, so we apply the product rule:

$\frac{d}{dx}\left( - e^y \sin(x) \right) = - \left( e^y \frac{d}{dx}(\sin(x)) + \sin(x) \frac{d}{dx}(e^y) \right)$

$= - \left( e^y \cos(x) + \sin(x) \cdot e^y \frac{dy}{dx} \right)$

Thus, the derivative of the left-hand side is:

$- e^y \cos(x) - e^y \sin(x) \frac{dy}{dx}$

Step 2: Differentiate the right-hand side

$\frac{d}{dx}(xy - 5y)$

This involves differentiating two terms:

1. $\frac{d}{dx}(xy)$ using the product rule: $\frac{d}{dx}(xy) = y + x \frac{dy}{dx}$

2. $\frac{d}{dx}(-5y)$: $\frac{d}{dx}(-5y) = -5 \frac{dy}{dx}$

So, the derivative of the right-hand side is:

$y + x \frac{dy}{dx} - 5 \frac{dy}{dx}$

Step 3: Combine both sides

Now we combine the derivatives of both sides:

$- e^y \cos(x) - e^y \sin(x) \frac{dy}{dx} = y + x \frac{dy}{dx} - 5 \frac{dy}{dx}$

Step 4: Solve for $\frac{dy}{dx}$

Collect all terms involving $\frac{dy}{dx}$ on one side:

$- e^y \sin(x) \frac{dy}{dx} - x \frac{dy}{dx} + 5 \frac{dy}{dx} = y + e^y \cos(x)$

Factor out $\frac{dy}{dx}$:

$\frac{dy}{dx} \left( - e^y \sin(x) - x + 5 \right) = y + e^y \cos(x)$

Finally, solve for $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{y + e^y \cos(x)}{- e^y \sin(x) - x + 5}$

This is the derivative $\frac{dy}{dx}$.

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Would you like more details on any part of the process?

Related Questions:

1. What is the product rule in differentiation?
2. How do we handle implicit differentiation in multivariable contexts?
3. Can we simplify the derivative further under specific conditions for $x$ and $y$?
4. What are the conditions where the denominator might be zero in this derivative?
5. How would the derivative change if we had a different trigonometric function?

Tip: Always group terms involving the derivative when using implicit differentiation to make solving for $\frac{dy}{dx}$ easier.