Let `f(x)=e^(x),g(x)={:{(x^(2),if,xlt(1)/(2)),(x-(1)/(4),if,xge(1)/(2)):}` and `h(x)=f(g(x))`. The derivative of `h(x)` and `x=(1)/(2)` is `e^((1)/(a))` then a equal to

Explanation

To solve the problem step by step, we will analyze the functions given and find the derivative of \( h(x) \) at \( x = \frac{1}{2} \).

### Step 1: Define the functions
We have:
- \( f(x) = e^x \)
- \( g(x) = \begin{cases} x^2 & \text{if } x < \frac{1}{2} \\ x - \frac{1}{4} & \text{if } x \geq \frac{1}{2} \end{cases} \)

### Step 2: Define \( h(x) \)
The function \( h(x) \) is defined as:
\[ h(x) = f(g(x)) \]
This means:
- For \( x < \frac{1}{2} \), \( h(x) = f(g(x)) = f(x^2) = e^{x^2} \)
- For \( x \geq \frac{1}{2} \), \( h(x) = f(g(x)) = f\left(x - \frac{1}{4}\right) = e^{x - \frac{1}{4}} \)

### Step 3: Evaluate \( h\left(\frac{1}{2}\right) \)
Since \( \frac{1}{2} \) falls in the second case (where \( x \geq \frac{1}{2} \)):
\[ h\left(\frac{1}{2}\right) = e^{\frac{1}{2} - \frac{1}{4}} = e^{\frac{1}{4}} \]

### Step 4: Find the derivative \( h'(x) \)
Now, we need to find the derivative \( h'(x) \):
- For \( x < \frac{1}{2} \):
\[ h'(x) = \frac{d}{dx}(e^{x^2}) = e^{x^2} \cdot 2x \]
- For \( x \geq \frac{1}{2} \):
\[ h'(x) = \frac{d}{dx}(e^{x - \frac{1}{4}}) = e^{x - \frac{1}{4}} \cdot 1 = e^{x - \frac{1}{4}} \]

### Step 5: Evaluate \( h'\left(\frac{1}{2}\right) \)
Using the second case for \( x \geq \frac{1}{2} \):
\[ h'\left(\frac{1}{2}\right) = e^{\frac{1}{2} - \frac{1}{4}} = e^{\frac{1}{4}} \]

### Step 6: Set the derivative equal to the given value
We are given that:
\[ h'\left(\frac{1}{2}\right) = e^{\frac{1}{a}} \]
From our calculation, we found:
\[ e^{\frac{1}{4}} = e^{\frac{1}{a}} \]

### Step 7: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ \frac{1}{4} = \frac{1}{a} \]

### Step 8: Solve for \( a \)
Cross-multiplying gives:
\[ a = 4 \]

Thus, the value of \( a \) is \( 4 \).

### Final Answer
The value of \( a \) is \( 4 \).

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