Let f and g be differentiable functions satisfying `g(a) = b,g' (a) = 2` and fog =I (identity function). then f' (b) is equal to

Options

A. 2
B. `(2)/(3)`
C. `(1)/(2)`
D. None of these

Answer

To solve the problem, we need to find \( f'(b) \) given the conditions \( g(a) = b \), \( g'(a) = 2 \), and \( f \circ g = I \) (the identity function).

### Step-by-Step Solution:

1. **Understanding the Identity Function**:
Since \( f(g(x)) = x \) for all \( x \), this means that \( f \) is the inverse function of \( g \).

2. **Differentiating Both Sides**:
We differentiate \( f(g(x)) = x \) with respect to \( x \) using the chain rule:
\[ \frac{d}{dx}[f(g(x))] = \frac{d}{dx}[x] \]
This gives us:
\[ f'(g(x)) \cdot g'(x) = 1 \]

3. **Substituting \( x = a \)**:
We substitute \( x = a \) into the differentiated equation:
\[ f'(g(a)) \cdot g'(a) = 1 \]

4. **Using Given Values**:
From the problem, we know that \( g(a) = b \) and \( g'(a) = 2 \). Substituting these values into the equation gives:
\[ f'(b) \cdot 2 = 1 \]

5. **Solving for \( f'(b) \)**:
To isolate \( f'(b) \), we divide both sides by 2:
\[ f'(b) = \frac{1}{2} \]

### Final Answer:
Thus, \( f'(b) = \frac{1}{2} \).

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