Differentiate `sqrt(sin(e^(x)))` w.r.t. x.

Answer

To differentiate the function \( y = \sqrt{\sin(e^x)} \) with respect to \( x \), we will use the chain rule and the derivatives of the functions involved. Here’s the step-by-step solution:

### Step 1: Rewrite the function
Let \( y = \sqrt{\sin(e^x)} \).

### Step 2: Apply the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. The derivative of \( \sqrt{u} \) where \( u = \sin(e^x) \) is given by:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{u}} \cdot \frac{du}{dx} \]
where \( u = \sin(e^x) \).

### Step 3: Differentiate \( u = \sin(e^x) \)
Now, we need to find \( \frac{du}{dx} \):
\[ \frac{du}{dx} = \cos(e^x) \cdot \frac{d}{dx}(e^x) \]
The derivative of \( e^x \) is \( e^x \), so:
\[ \frac{du}{dx} = \cos(e^x) \cdot e^x \]

### Step 4: Substitute back into the derivative
Now we substitute \( u \) and \( \frac{du}{dx} \) back into the derivative of \( y \):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{\sin(e^x)}} \cdot (\cos(e^x) \cdot e^x) \]

### Step 5: Simplify the expression
Thus, the derivative can be expressed as:
\[ \frac{dy}{dx} = \frac{e^x \cos(e^x)}{2\sqrt{\sin(e^x)}} \]

### Final Answer
The derivative of \( y = \sqrt{\sin(e^x)} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{e^x \cos(e^x)}{2\sqrt{\sin(e^x)}} \]

---