(1 point) Consider the equation −𝑒𝑦sin(𝑥)=𝑥𝑦−5𝑦
. Find 𝑑𝑦𝑑𝑥
.


𝑑𝑦𝑑𝑥=

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 \(\frac{dy}{dx}\).

---

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.

Consider the equation −e^y sin(x) = xy - 5y. Find dy/dx.

Learn how to solve the implicit differentiation problem −e^y sin(x) = xy - 5y to find dy/dx. This step-by-step solution covers the product rule and implicit differentiation methods. Suitable for calculus students in Grades 11-12.

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