`d/dx(sqrt(e^(sqrtx)))` = ___________.

Answer

To solve the problem \( \frac{d}{dx} \left( \sqrt{e^{\sqrt{x}}} \right) \), we will use the chain rule and the properties of derivatives. Here’s a step-by-step solution:

### Step 1: Rewrite the expression
We start by rewriting the square root in exponential form:
\[ \sqrt{e^{\sqrt{x}}} = (e^{\sqrt{x}})^{1/2} = e^{\frac{1}{2} \sqrt{x}} \]

### Step 2: Differentiate using the chain rule
Now, we differentiate \( e^{\frac{1}{2} \sqrt{x}} \):
\[ \frac{d}{dx} \left( e^{\frac{1}{2} \sqrt{x}} \right) = e^{\frac{1}{2} \sqrt{x}} \cdot \frac{d}{dx} \left( \frac{1}{2} \sqrt{x} \right) \]

### Step 3: Differentiate \( \frac{1}{2} \sqrt{x} \)
Next, we differentiate \( \frac{1}{2} \sqrt{x} \):
\[ \frac{d}{dx} \left( \frac{1}{2} \sqrt{x} \right) = \frac{1}{2} \cdot \frac{1}{2\sqrt{x}} = \frac{1}{4\sqrt{x}} \]

### Step 4: Substitute back into the derivative
Now we substitute this result back into our derivative:
\[ \frac{d}{dx} \left( e^{\frac{1}{2} \sqrt{x}} \right) = e^{\frac{1}{2} \sqrt{x}} \cdot \frac{1}{4\sqrt{x}} \]

### Step 5: Final expression
Thus, the final result for the derivative is:
\[ \frac{d}{dx} \left( \sqrt{e^{\sqrt{x}}} \right) = \frac{1}{4\sqrt{x}} e^{\frac{1}{2} \sqrt{x}} \]

### Final Answer:
\[ \frac{d}{dx} \left( \sqrt{e^{\sqrt{x}}} \right) = \frac{1}{4\sqrt{x}} e^{\frac{1}{2} \sqrt{x}} \]

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