if ` y= x +e^x ,` then ` (d^2 x)/(dy^2)` is

Options

A. `e^x`
B. `-(e^x)/((1+e^x )^3)`
C. `-(e^x)/((1+e^x)^2)`
D. `(1)/((1+e^x)^2)`

Answer

To find \( \frac{d^2 x}{dy^2} \) when \( y = x + e^x \), we will follow these steps:

### Step 1: Differentiate \( y \) with respect to \( x \)
Given:
\[ y = x + e^x \]
Differentiating both sides with respect to \( x \):
\[ \frac{dy}{dx} = 1 + e^x \]

### Step 2: Find \( \frac{dx}{dy} \)
To find \( \frac{dx}{dy} \), we take the reciprocal of \( \frac{dy}{dx} \):
\[ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{1 + e^x} \]

### Step 3: Differentiate \( \frac{dx}{dy} \) with respect to \( y \) to find \( \frac{d^2 x}{dy^2} \)
Now we need to differentiate \( \frac{dx}{dy} \) with respect to \( y \):
\[ \frac{d^2 x}{dy^2} = \frac{d}{dy}\left(\frac{dx}{dy}\right) = \frac{d}{dy}\left(\frac{1}{1 + e^x}\right) \]
Using the quotient rule:
\[ \frac{d}{dy}\left(\frac{1}{1 + e^x}\right) = \frac{0 \cdot (1 + e^x) - 1 \cdot \frac{de^x}{dy}}{(1 + e^x)^2} \]
Now, we need to find \( \frac{de^x}{dy} \):
\[ \frac{de^x}{dy} = \frac{de^x}{dx} \cdot \frac{dx}{dy} = e^x \cdot \frac{1}{1 + e^x} \]
Substituting this back into our expression:
\[ \frac{d^2 x}{dy^2} = \frac{-e^x \cdot \frac{1}{1 + e^x}}{(1 + e^x)^2} \]
This simplifies to:
\[ \frac{d^2 x}{dy^2} = \frac{-e^x}{(1 + e^x)^3} \]

### Final Answer
Thus, the final result is:
\[ \frac{d^2 x}{dy^2} = \frac{-e^x}{(1 + e^x)^3} \]