Understanding Second Derivative of Inverse Function

This question asks us to find the expression for the second derivative of x with respect to y, denoted as \(\dfrac{d^2x}{dy^2}\), using the derivatives of y with respect to x (\(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\)). This involves working with inverse functions and applying differentiation rules.

Relating Derivatives: The Inverse Function Rule

We start with the fundamental relationship between the first derivatives of a function and its inverse. If \(y\) is a function of \(x\), then \(x\) is a function of \(y\). The rule for the first derivative of the inverse function is:

\(\dfrac{dx}{dy} = \dfrac{1}{\frac{dy}{dx}}\)

This can also be written using negative exponents:

\(\dfrac{dx}{dy} = \left(\dfrac{dy}{dx}\right)^{-1}\)

Calculating the Second Derivative

To find \(\dfrac{d^2x}{dy^2}\), we need to differentiate \(\dfrac{dx}{dy}\) with respect to \(y\). We will use the chain rule extensively here. Remember that \(\dfrac{d}{dy} = \dfrac{dx}{dy} \dfrac{d}{dx}\).

1. Differentiate \(\dfrac{dx}{dy}\) with respect to \(y\):

   \(\dfrac{d^2x}{dy^2} = \dfrac{d}{dy}\left(\dfrac{dx}{dy}\right)\)

2. Substitute the expression for \(\dfrac{dx}{dy}\):

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

3. Apply the chain rule \(\dfrac{d}{dy} = \dfrac{dx}{dy} \dfrac{d}{dx}\):

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

4. Substitute \(\dfrac{dx}{dy} = \left(\dfrac{dy}{dx}\right)^{-1}\) again:

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

5. Now, differentiate the term \(\left(\dfrac{dy}{dx}\right)^{-1}\) with respect to \(x\). Using the power rule and chain rule:

   \(\dfrac{d}{dx}\left(\left(\dfrac{dy}{dx}\right)^{-1}\right) = -1 \cdot \left(\dfrac{dy}{dx}\right)^{-2} \cdot \dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)\)

   \(= -\left(\dfrac{dy}{dx}\right)^{-2} \cdot \dfrac{d^2y}{dx^2}\)

6. Substitute this result back into the expression for \(\dfrac{d^2x}{dy^2}\):

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

7. Combine the terms:

   \(\dfrac{d^2x}{dy^2} = -\left(\dfrac{dy}{dx}\right)^{-1} \left(\dfrac{dy}{dx}\right)^{-2} \dfrac{d^2y}{dx^2}\)

   \(\dfrac{d^2x}{dy^2} = -\left(\dfrac{dy}{dx}\right)^{-3} \dfrac{d^2y}{dx^2}\)

8. Rewrite the expression in the format matching the options:

   \(\dfrac{d^2x}{dy^2} = -\dfrac{d^2y}{dx^2} \left(\dfrac{dy}{dx}\right)^{-3}\)

Comparing with Options

The derived expression matches option 4.

Final Answer Check

The calculation confirms that \(\dfrac{d^2x}{dy^2}\) is equivalent to \(-\dfrac{d^2y}{dx^2}\left(\dfrac{dy}{dx}\right)^{-3}\). This formula is crucial when dealing with the second derivative of inverse functions in calculus.