for any differential function ` y= F` (x) : the value of ` ( d^2 y) /( dx^2) +((dy)/(dx)) ^3 . (d^2 x)/( dy^2)`

Options

A. 1
B. 2
C. 0
D. 4

Answer

To solve the given expression \( \frac{d^2 y}{dx^2} + \left( \frac{dy}{dx} \right)^3 \cdot \frac{d^2 x}{dy^2} \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Understanding the Terms**:
- We have \( y = F(x) \), which means \( y \) is a function of \( x \).
- We need to find the second derivative of \( y \) with respect to \( x \) and the second derivative of \( x \) with respect to \( y \).

2. **Finding \( \frac{d^2 x}{dy^2} \)**:
- We know that \( \frac{d^2 x}{dy^2} = \frac{d}{dy}\left(\frac{dx}{dy}\right) \).
- Using the chain rule, we can express this as:
\[ \frac{d^2 x}{dy^2} = \frac{d}{dy}\left(\frac{1}{\frac{dy}{dx}}\right) = -\frac{1}{\left(\frac{dy}{dx}\right)^2} \cdot \frac{d^2y}{dx^2} \]

3. **Substituting into the Expression**:
- Now we substitute \( \frac{d^2 x}{dy^2} \) into the original expression:
\[ \frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^3 \cdot \left(-\frac{1}{\left(\frac{dy}{dx}\right)^2} \cdot \frac{d^2 y}{dx^2}\right) \]

4. **Simplifying the Expression**:
- This simplifies to:
\[ \frac{d^2 y}{dx^2} - \frac{dy}{dx} \cdot \frac{d^2 y}{dx^2} \]
- Factoring out \( \frac{d^2 y}{dx^2} \):
\[ \frac{d^2 y}{dx^2} \left(1 - \frac{dy}{dx}\right) \]

5. **Final Result**:
- Therefore, the final value of the expression is:
\[ \frac{d^2 y}{dx^2} \left(1 - \frac{dy}{dx}\right) \]