If `y=e^(2x) ` then ` (d^2 y)/(dx^2) . (d^2 x)/(dy^2)` is equal to

Options

A. `e^(-2x)`
B. `-2e^(-2x)`
C. `2e^(-2x)`
D. `1`

Answer

To solve the problem, we need to find the expression \( \frac{d^2 y}{dx^2} \cdot \frac{d^2 x}{dy^2} \) given that \( y = e^{2x} \).

### Step-by-Step Solution:

1. **Find \( \frac{dy}{dx} \)**:
\[ y = e^{2x} \]
Using the chain rule, we differentiate:
\[ \frac{dy}{dx} = e^{2x} \cdot \frac{d}{dx}(2x) = e^{2x} \cdot 2 = 2e^{2x} \]

2. **Find \( \frac{d^2y}{dx^2} \)**:
Now, we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(2e^{2x}) = 2 \cdot e^{2x} \cdot \frac{d}{dx}(2x) = 2 \cdot e^{2x} \cdot 2 = 4e^{2x} \]

3. **Find \( x \) in terms of \( y \)**:
From \( y = e^{2x} \), we can take the natural logarithm:
\[ \ln(y) = 2x \implies x = \frac{1}{2} \ln(y) \]

4. **Find \( \frac{dx}{dy} \)**:
Differentiate \( x \) with respect to \( y \):
\[ \frac{dx}{dy} = \frac{1}{2} \cdot \frac{1}{y} \]

5. **Find \( \frac{d^2x}{dy^2} \)**:
Differentiate \( \frac{dx}{dy} \) again:
\[ \frac{d^2x}{dy^2} = \frac{d}{dy}\left(\frac{1}{2y}\right) = -\frac{1}{2y^2} \]

6. **Combine the results**:
Now we can find the product:
\[ \frac{d^2y}{dx^2} \cdot \frac{d^2x}{dy^2} = (4e^{2x}) \cdot \left(-\frac{1}{2y^2}\right) \]
Substitute \( y = e^{2x} \):
\[ = 4e^{2x} \cdot \left(-\frac{1}{2(e^{2x})^2}\right) = 4e^{2x} \cdot \left(-\frac{1}{2e^{4x}}\right) \]
Simplifying this:
\[ = -\frac{4}{2} \cdot \frac{e^{2x}}{e^{4x}} = -2 \cdot e^{-2x} \]

### Final Result:
Thus, the final answer is:
\[ \frac{d^2 y}{dx^2} \cdot \frac{d^2 x}{dy^2} = -2e^{-2x} \]