If `e^(xy)-4xy=2," then "(dy)/(dx)=`

Options

A. `(y)/(x)`
B. `(x)/(y)`
C. `(-x)/(y)`
D. `(-y)/(x)`

Answer

To solve the equation \( e^{xy} - 4xy = 2 \) for \( \frac{dy}{dx} \), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Rewrite the equation
We start with the equation:
\[ e^{xy} - 4xy = 2 \]
To differentiate, we can rewrite it as:
\[ e^{xy} - 4xy - 2 = 0 \]

### Step 2: Differentiate both sides with respect to \( x \)
Now, we differentiate both sides of the equation with respect to \( x \):
\[ \frac{d}{dx}(e^{xy}) - \frac{d}{dx}(4xy) - \frac{d}{dx}(2) = 0 \]

### Step 3: Apply the chain rule to \( e^{xy} \)
Using the chain rule, the derivative of \( e^{xy} \) is:
\[ \frac{d}{dx}(e^{xy}) = e^{xy} \cdot \frac{d}{dx}(xy) \]
Now, we apply the product rule to differentiate \( xy \):
\[ \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \]
Thus,
\[ \frac{d}{dx}(e^{xy}) = e^{xy} \cdot (x \frac{dy}{dx} + y) \]

### Step 4: Differentiate \( 4xy \)
Now, we differentiate \( 4xy \):
\[ \frac{d}{dx}(4xy) = 4 \left( x \frac{dy}{dx} + y \right) \]

### Step 5: Substitute derivatives back into the equation
Substituting back into our differentiated equation gives:
\[ e^{xy} (x \frac{dy}{dx} + y) - 4(x \frac{dy}{dx} + y) = 0 \]

### Step 6: Rearranging the equation
We can rearrange the equation:
\[ e^{xy} (x \frac{dy}{dx} + y) = 4(x \frac{dy}{dx} + y) \]

### Step 7: Factor out common terms
Now, we can factor out \( (x \frac{dy}{dx} + y) \):
\[ (e^{xy} - 4)(x \frac{dy}{dx} + y) = 0 \]
Since \( e^{xy} - 4 \) cannot be zero (as it would contradict our original equation), we have:
\[ x \frac{dy}{dx} + y = 0 \]

### Step 8: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \):
\[ x \frac{dy}{dx} = -y \]
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{y}{x} \]

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