f and g are two differentiable functions which satisfy the condition ` g'(a) = 2,g (a) =b and (f og) =I` identity function , then f'(b) is equal to

Options

A. `2/3`
B. `1/2`
C. `2`
D. None

Answer

To solve the problem, we need to find \( f'(b) \) given the conditions provided. Let's break it down step by step.

### Step 1: Understand the Identity Function
The identity function is defined as \( I(x) = x \). This means that for any input \( x \), the output is the same, i.e., \( f(g(x)) = x \).

### Step 2: Differentiate the Identity Function
Since \( f(g(x)) = x \), we can differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}[f(g(x))] = \frac{d}{dx}[x] \]
Using the chain rule on the left side, we have:
\[ f'(g(x)) \cdot g'(x) = 1 \]

### Step 3: Substitute the Given Values
We know from the problem statement:
- \( g'(a) = 2 \)
- \( g(a) = b \)

Now, we can substitute \( x = a \) into our differentiated equation:
\[ f'(g(a)) \cdot g'(a) = 1 \]
Substituting \( g(a) = b \) and \( g'(a) = 2 \):
\[ f'(b) \cdot 2 = 1 \]

### Step 4: Solve for \( f'(b) \)
Now, we can solve for \( f'(b) \):
\[ f'(b) = \frac{1}{2} \]

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

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