The function `f(x)=e^(x)+x,` being differentiable and one- to -one, has a differentiable inverse `f^(-1)(x).` The value of `(x)/(dx)(f^(-1)at f(log2)` is

Options

A. `(1)/(In2)`
B. `(1)/(3)`
C. `(1)/(4)`
D. none of these

Explanation

To find the value of \(\frac{dx}{d(f^{-1}(x))}\) at \(f(\log 2)\), we can follow these steps:

### Step 1: Understand the relationship between \(f\) and \(f^{-1}\)
Given that \(f(x) = e^x + x\), we know that \(f\) is a one-to-one function, which means it has an inverse \(f^{-1}(x)\).

### Step 2: Use the formula for the derivative of the inverse function
The derivative of the inverse function can be expressed as:
\[
...