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The image contains a differential equation problem with the equation:

\[ e^{-xy} - 4xy = 0 \]

It asks for the value of \(\frac{dy}{dx}\), and the choices are:

a) \(\frac{y}{x}\)

b) \(\frac{y}{x}\)

c) \(\frac{x}{y}\)

d) none of these

Let me solve this step by step to find the correct answer.

### Step 1: Solve for \( e^{-xy} \)

From the given equation:

\[
e^{-xy} = 4xy
\]

### Step 2: Implicit differentiation

To find \(\frac{dy}{dx}\), we need to differentiate both sides of the equation implicitly with respect to \(x\). First, applying the product and chain rule to \( e^{-xy} \):

\[
\frac{d}{dx}(e^{-xy}) = e^{-xy} \cdot \frac{d}{dx}(-xy)
\]
\[
= e^{-xy} \cdot (-y - x \frac{dy}{dx})
\]

Next, differentiate the right-hand side \( 4xy \):

\[
\frac{d}{dx}(4xy) = 4 \left( y + x \frac{dy}{dx} \right)
\]

### Step 3: Set up the equation

Now equate the two differentiated expressions:

\[
e^{-xy} \cdot (-y - x \frac{dy}{dx}) = 4(y + x \frac{dy}{dx})
\]

### Step 4: Substitute \(e^{-xy} = 4xy\)

Substitute \( e^{-xy} = 4xy \) into the equation:

\[
4xy(-y - x \frac{dy}{dx}) = 4(y + x \frac{dy}{dx})
\]

### Step 5: Simplify

Cancel the common factor of 4 on both sides:

\[
xy(-y - x \frac{dy}{dx}) = y + x \frac{dy}{dx}
\]

Expand both sides:

\[
-xy^2 - x^2y \frac{dy}{dx} = y + x \frac{dy}{dx}
\]

### Step 6: Collect terms involving \(\frac{dy}{dx}\)

Bring all terms involving \(\frac{dy}{dx}\) to one side:

\[
- x^2y \frac{dy}{dx} - x \frac{dy}{dx} = y + xy^2
\]

Factor out \(\frac{dy}{dx}\):

\[
\frac{dy}{dx}(-x^2y - x) = y + xy^2
\]

### Step 7: Solve for \(\frac{dy}{dx}\)

Finally, solve for \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = \frac{y + xy^2}{-x^2y - x}
\]

This expression does not simplify to any of the given options, so the correct answer is **d) none of these**.

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:

1. What is implicit differentiation and when is it used?
2. How does the chain rule apply to exponential functions like \( e^{-xy} \)?
3. What is the significance of product rule in this context?
4. How can you determine whether a differential equation is separable or not?
5. What are some common mistakes to avoid when solving implicit differentiation problems?

**Tip:** Always check if a term can be factored out to simplify the expression for \(\frac{dy}{dx}\). This helps in reducing complex terms during differentiation.

Given e^-xy - 4xy = 0, find the value of dy/dx.

Learn how to solve the differential equation e^-xy - 4xy = 0 using implicit differentiation. This problem involves applying the product and chain rule to exponential functions, leading to a step-by-step solution. Suitable for college-level students studying differential equations.

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