If `sqrt(xy)=1," then "(dy)/(dx)=`

Options

A. `(-1)/(x^(2))`
B. `(1)/(x^(2))`
C. `-x^(2)`
D. `x^(2)`

Answer

To solve the problem where \( \sqrt{xy} = 1 \) and find \( \frac{dy}{dx} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sqrt{xy} = 1 \]
Squaring both sides gives:
\[ xy = 1 \]

### Step 2: Differentiate both sides
Now we will differentiate both sides with respect to \( x \). Using the product rule on the left side, we have:
\[ \frac{d}{dx}(xy) = \frac{d}{dx}(1) \]
The right side differentiates to 0:
\[ \frac{d}{dx}(1) = 0 \]

### Step 3: Apply the product rule
Using the product rule \( \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \), we differentiate \( xy \):
\[ x \frac{dy}{dx} + y \frac{dx}{dx} = 0 \]
Since \( \frac{dx}{dx} = 1 \), we can simplify this to:
\[ x \frac{dy}{dx} + y = 0 \]

### Step 4: Solve for \( \frac{dy}{dx} \)
Rearranging the equation gives:
\[ x \frac{dy}{dx} = -y \]
Dividing both sides by \( x \):
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Step 5: Substitute \( y \) in terms of \( x \)
From our earlier equation \( xy = 1 \), we can express \( y \) as:
\[ y = \frac{1}{x} \]
Substituting this into the equation for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -\frac{\frac{1}{x}}{x} = -\frac{1}{x^2} \]

### Final Answer
Thus, we find that:
\[ \frac{dy}{dx} = -\frac{1}{x^2} \]

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