Differentiate the following w.r.t. x :
`sqrt(e^(sqrtx)),xgt0`

Answer

To differentiate the function \( y = \sqrt{e^{\sqrt{x}}} \) with respect to \( x \) where \( x > 0 \), we will follow these steps:

### Step 1: Rewrite the function
We start by rewriting the function in a more manageable form:
\[ y = (e^{\sqrt{x}})^{1/2} = e^{\frac{1}{2} \sqrt{x}} \]

### Step 2: Differentiate using the chain rule
Now we differentiate \( y \) with respect to \( x \). We will use the chain rule for differentiation. The derivative of \( e^{u} \) with respect to \( x \) is \( e^{u} \cdot \frac{du}{dx} \), where \( u = \frac{1}{2} \sqrt{x} \).

First, we find \( \frac{du}{dx} \):
\[ u = \frac{1}{2} \sqrt{x} \implies \frac{du}{dx} = \frac{1}{2} \cdot \frac{1}{2\sqrt{x}} = \frac{1}{4\sqrt{x}} \]

Now we can differentiate \( y \):
\[ \frac{dy}{dx} = e^{\frac{1}{2} \sqrt{x}} \cdot \frac{du}{dx} = e^{\frac{1}{2} \sqrt{x}} \cdot \frac{1}{4\sqrt{x}} \]

### Step 3: Simplify the expression
Now we simplify the expression:
\[ \frac{dy}{dx} = \frac{1}{4\sqrt{x}} e^{\frac{1}{2} \sqrt{x}} \]

### Final Result
Thus, the derivative of \( y = \sqrt{e^{\sqrt{x}}} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{4\sqrt{x}} e^{\frac{1}{2} \sqrt{x}} \]