If `sqrt(xy) =1`, then `(d^(2)y)/(dx^(2))=`

Options

A. `2x^(3)`
B. `-2x^(3)`
C. `-2/x^(3)`
D. `2/x^(3)`

Answer

To solve the problem step by step, we start with the given equation and apply differentiation techniques.

### Step 1: Start with the given equation
We are given:
\[ \sqrt{xy} = 1 \]

### Step 2: Square both sides
To eliminate the square root, we square both sides:
\[ xy = 1 \]

### Step 3: Express \(y\) in terms of \(x\)
From the equation \(xy = 1\), we can express \(y\) as:
\[ y = \frac{1}{x} \]

### Step 4: Differentiate \(y\) with respect to \(x\)
Now, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{1}{x}\right) \]
Using the power rule, we rewrite \(\frac{1}{x}\) as \(x^{-1}\):
\[ \frac{dy}{dx} = -\frac{1}{x^2} \]

### Step 5: Differentiate again to find \(\frac{d^2y}{dx^2}\)
Now, we differentiate \(\frac{dy}{dx}\) again with respect to \(x\):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(-\frac{1}{x^2}\right) \]
Again using the power rule, we rewrite \(-\frac{1}{x^2}\) as \(-x^{-2}\):
\[ \frac{d^2y}{dx^2} = 2x^{-3} = \frac{2}{x^3} \]

### Final Result
Thus, the second derivative of \(y\) with respect to \(x\) is:
\[ \frac{d^2y}{dx^2} = \frac{2}{x^3} \]

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