If `f(x)=(1)/(1-x)`, then the derivative of the composite function f[f{f(x)}]` is equal to

Options

A. 0
B. `1//2`
C. 1
D. 2

Explanation

To find the derivative of the composite function \( g(x) = f(f(f(x))) \) where \( f(x) = \frac{1}{1-x} \), we will follow these steps:

### Step 1: Find \( f(f(x)) \)

We start by calculating \( f(f(x)) \).

\[ f(x) = \frac{1}{1-x} \]

Now, substituting \( f(x) \) into itself:

\[ f(f(x)) = f\left(\frac{1}{1-x}\right) = \frac{1}{1 - \frac{1}{1-x}} \]

To simplify this, we need to combine the terms in the denominator:

\[ 1 - \frac{1}{1-x} = \frac{(1-x) - 1}{1-x} = \frac{-x}{1-x} \]

Thus,

\[ f(f(x)) = \frac{1}{\frac{-x}{1-x}} = -\frac{1-x}{x} = \frac{x-1}{x} \]

### Step 2: Find \( f(f(f(x))) \)

Next, we find \( f(f(f(x))) \):

\[ f(f(f(x))) = f\left(\frac{x-1}{x}\right) = \frac{1}{1 - \frac{x-1}{x}} \]

Again, simplifying the denominator:

\[ 1 - \frac{x-1}{x} = \frac{x - (x-1)}{x} = \frac{1}{x} \]

Thus,

\[ f(f(f(x))) = \frac{1}{\frac{1}{x}} = x \]

### Step 3: Find the derivative \( g'(x) \)

Now that we have \( g(x) = f(f(f(x))) = x \), we can find the derivative:

\[ g'(x) = \frac{d}{dx}(x) = 1 \]

### Final Answer

The derivative of the composite function \( f(f(f(x))) \) is:

\[ g'(x) = 1 \]

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