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 \((\frac{d^2y}{dx^2}) \cdot (\frac{d^2x}{dy^2})\) given that \(y = e^{2x}\).

### Step 1: Find \(\frac{dy}{dx}\)
Given \(y = e^{2x}\), we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}(e^{2x}) = 2e^{2x} \]

**Hint:** Use the chain rule for differentiation when the exponent is a function of \(x\).

### Step 2: Find \(\frac{d^2y}{dx^2}\)
Next, we differentiate \(\frac{dy}{dx}\) to find \(\frac{d^2y}{dx^2}\):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(2e^{2x}) = 2 \cdot \frac{d}{dx}(e^{2x}) = 2 \cdot 2e^{2x} = 4e^{2x} \]

**Hint:** Differentiate again to find the second derivative, applying the chain rule once more.

### Step 3: Find \(\frac{dx}{dy}\)
To find \(\frac{dx}{dy}\), we use the relationship:
\[ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{2e^{2x}} \]

**Hint:** The derivative \(\frac{dx}{dy}\) is the reciprocal of \(\frac{dy}{dx}\).

### Step 4: Find \(\frac{d^2x}{dy^2}\)
Now, we differentiate \(\frac{dx}{dy}\) with respect to \(y\) to find \(\frac{d^2x}{dy^2}\):
\[ \frac{d^2x}{dy^2} = \frac{d}{dy}\left(\frac{1}{2e^{2x}}\right) \]
Using the chain rule:
\[ \frac{d^2x}{dy^2} = -\frac{1}{(2e^{2x})^2} \cdot \frac{d(2e^{2x})}{dy} \]
Now, since \(\frac{dy}{dx} = 2e^{2x}\), we can find \(\frac{d(2e^{2x})}{dy}\):
\[ \frac{d(2e^{2x})}{dy} = \frac{d(2e^{2x})}{dx} \cdot \frac{dx}{dy} = 4e^{2x} \cdot \frac{1}{2e^{2x}} = 2 \]
Thus,
\[ \frac{d^2x}{dy^2} = -\frac{1}{(2e^{2x})^2} \cdot 2 = -\frac{2}{4e^{4x}} = -\frac{1}{2e^{4x}} \]

**Hint:** When differentiating with respect to \(y\), remember to use the chain rule and the reciprocal relationship between derivatives.

### Step 5: Calculate \((\frac{d^2y}{dx^2}) \cdot (\frac{d^2x}{dy^2})\)
Now we can substitute our results into the expression:
\[ \frac{d^2y}{dx^2} \cdot \frac{d^2x}{dy^2} = (4e^{2x}) \cdot \left(-\frac{1}{2e^{4x}}\right) \]
Simplifying this gives:
\[ = -\frac{4e^{2x}}{2e^{4x}} = -\frac{4}{2e^{2x}} = -\frac{2}{e^{2x}} \]

### Final Answer:
\[ \frac{d^2y}{dx^2} \cdot \frac{d^2x}{dy^2} = -\frac{2}{e^{2x}} \]