`(d^2 x)/(dy^2)` is equal to :

Options

A. `-((d^(2)y)/(dx^(2)))^(-1)((dy)/(dx))^(-30`
B. `((d^(2)y)/(dx^(2)))((dy)/(dx))^(-2)`
C. `-((d^(2)y)/(dx^(2)))((dy)/(dx))^(-3)`
D. `((d^(2)y)/(dx^(2)))^(-1)`

Answer

To solve the problem of finding \(\frac{d^2 x}{dy^2}\), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Start with the relationship**: We know that \(\frac{dx}{dy} = \left(\frac{dy}{dx}\right)^{-1}\). This means that the derivative of \(x\) with respect to \(y\) is the reciprocal of the derivative of \(y\) with respect to \(x\).

\[ \frac{dx}{dy} = \left(\frac{dy}{dx}\right)^{-1} \]

**Hint**: Remember that the derivative of one variable with respect to another can often be expressed as the reciprocal of the derivative in the opposite direction.

2. **Differentiate both sides with respect to \(y\)**: We will differentiate \(\frac{dx}{dy}\) with respect to \(y\).

\[ \frac{d^2x}{dy^2} = \frac{d}{dy}\left(\left(\frac{dy}{dx}\right)^{-1}\right) \]

**Hint**: When differentiating a function that is raised to a power, use the chain rule.

3. **Apply the chain rule**: Using the chain rule, we differentiate \(\left(\frac{dy}{dx}\right)^{-1}\).

\[ \frac{d^2x}{dy^2} = -\left(\frac{dy}{dx}\right)^{-2} \cdot \frac{d}{dy}\left(\frac{dy}{dx}\right) \]

**Hint**: The derivative of \(u^{-1}\) is \(-u^{-2} \cdot \frac{du}{dy}\).

4. **Express \(\frac{d}{dy}\left(\frac{dy}{dx}\right)\)**: We can express \(\frac{d}{dy}\left(\frac{dy}{dx}\right)\) using the chain rule again.

\[ \frac{d}{dy}\left(\frac{dy}{dx}\right) = \frac{d^2y}{dx^2} \cdot \frac{dx}{dy} \]

**Hint**: When differentiating \(\frac{dy}{dx}\) with respect to \(y\), apply the chain rule to express it in terms of \(\frac{d^2y}{dx^2}\).

5. **Substitute back into the equation**: Now substitute this expression back into the equation for \(\frac{d^2x}{dy^2}\).

\[ \frac{d^2x}{dy^2} = -\left(\frac{dy}{dx}\right)^{-2} \cdot \left(\frac{d^2y}{dx^2} \cdot \frac{dx}{dy}\right) \]

6. **Simplify the expression**: We know that \(\frac{dx}{dy} = \left(\frac{dy}{dx}\right)^{-1}\), so we can substitute this into our equation.

\[ \frac{d^2x}{dy^2} = -\left(\frac{dy}{dx}\right)^{-2} \cdot \left(\frac{d^2y}{dx^2} \cdot \left(\frac{dy}{dx}\right)^{-1}\right) \]

This simplifies to:

\[ \frac{d^2x}{dy^2} = -\frac{d^2y}{dx^2} \cdot \left(\frac{dy}{dx}\right)^{-3} \]

**Hint**: When simplifying, pay attention to how powers of the same base can be combined.

7. **Final Result**: Thus, we have:

\[ \frac{d^2x}{dy^2} = -\frac{d^2y}{dx^2} \cdot \left(\frac{dy}{dx}\right)^{-3} \]

### Conclusion:
The final expression for \(\frac{d^2x}{dy^2}\) is:

\[ \frac{d^2x}{dy^2} = -\frac{d^2y}{dx^2} \cdot \left(\frac{dy}{dx}\right)^{-3} \]