If `R=([1+((dy)/(dx))^2]^(-3//2))/((d^2y)/(dx2))` , then`R^(2//3)` can be put in the form of `1/(((d^2y)/(dx^2))^(2//3))+1/(((d^2x)/(dy^2))^(2//3))` b. `1/(((d^2y)/(dx^2))^(2//3))-1/(((d^2x)/(dy^2))^(2//3))` c. `2/(((d^2y)/(dx^2))^(2//3))+2/(((d^2x)/(dy^2))^(2//3))` d. `1/(((d^2y)/(dx^2))^(2//3))dot1/(((d^2x)/(dy^2))^(2//3))`

Options

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

Explanation

To solve the given problem, we start with the expression for \( R \):

\[ R = \frac{[1 + \left(\frac{dy}{dx}\right)^2]^{-\frac{3}{2}}}{\frac{d^2y}{dx^2}} \]

We need to find \( R^{\frac{2}{3}} \) and express it in a specific form.

### Step 1: Calculate \( R^{\frac{2}{3}} \)

First, we raise \( R \) to the power of \( \frac{2}{3} \):

\[ R^{\frac{2}{3}} = \left(\frac{[1 + \left(\frac{dy}{dx}\right)^2]^{-\frac{3}{2}}}{\frac{d^2y}{dx^2}}\right)^{\frac{2}{3}} \]

This can be simplified as:

\[ R^{\frac{2}{3}} = \frac{[1 + \left(\frac{dy}{dx}\right)^2]^{-1}}{\left(\frac{d^2y}{dx^2}\right)^{\frac{2}{3}}} \]

### Step 2: Simplify the expression

Next, we simplify the expression further:

\[ R^{\frac{2}{3}} = \frac{1}{[1 + \left(\frac{dy}{dx}\right)^2] \cdot \left(\frac{d^2y}{dx^2}\right)^{\frac{2}{3}}} \]

### Step 3: Find \( \frac{d^2x}{dy^2} \)

To relate this to \( \frac{d^2x}{dy^2} \), we need to find the second derivative \( \frac{d^2x}{dy^2} \). We know:

\[ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} \]

Differentiating this with respect to \( y \):

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

### Step 4: Substitute \( \frac{d^2x}{dy^2} \) into the expression

Now, substituting this into our expression for \( R^{\frac{2}{3}} \):

\[ R^{\frac{2}{3}} = \frac{1}{\left(\frac{d^2y}{dx^2}\right)^{\frac{2}{3}} + \left(-\frac{1}{\left(\frac{dy}{dx}\right)^2} \cdot \frac{d^2y}{dx^2}\right)^{\frac{2}{3}}} \]

### Step 5: Final expression

This leads us to:

\[ R^{\frac{2}{3}} = \frac{1}{\left(\frac{d^2y}{dx^2}\right)^{\frac{2}{3}} + \left(\frac{d^2x}{dy^2}\right)^{\frac{2}{3}}} \]

### Conclusion

Thus, we can conclude that:

\[ R^{\frac{2}{3}} = \frac{1}{\left(\frac{d^2y}{dx^2}\right)^{\frac{2}{3}} + \left(\frac{d^2x}{dy^2}\right)^{\frac{2}{3}}} \]

This matches option (a) from the question.