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 \( f[f[f(x)]] \) where \( f(x) = \frac{1}{1-x} \), we can follow these steps:

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

First, we need to compute \( f(f(x)) \).

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

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 find a common 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 compute \( f(f(f(x))) \):

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

Again, simplifying:

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

Thus,

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

### Step 3: Find the derivative \( f[f[f(x)]] \)

Now we have:

\[ g(x) = f(f(f(x))) = x \]

To find the derivative \( g'(x) \):

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

### Conclusion

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

\[ \boxed{1} \]

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