If `y=e^(sinsqrtx),"then "dy/dx` is :

Options

A. `e^(sinsqrtx)cossqrtx`
B. `(e^(sinsqrtx)cossqrtx)/(2sqrtx)`
C. `(e^(sinsqrtx))/(2sqrtx)`
D. None of these

Answer

To find the derivative of the function \( y = e^{\sin(\sqrt{x})} \), we will use the chain rule for differentiation. Let's go through the steps:

### Step 1: Differentiate the function
We start with the function:
\[ y = e^{\sin(\sqrt{x})} \]
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The derivative of \( e^u \) with respect to \( x \) is \( e^u \cdot \frac{du}{dx} \), where \( u = \sin(\sqrt{x}) \).

### Step 2: Differentiate the outer function
The outer function is \( e^{u} \), where \( u = \sin(\sqrt{x}) \). Thus, we have:
\[ \frac{dy}{dx} = e^{\sin(\sqrt{x})} \cdot \frac{d}{dx}(\sin(\sqrt{x})) \]

### Step 3: Differentiate the inner function
Next, we need to differentiate \( \sin(\sqrt{x}) \). Again, we apply the chain rule:
\[ \frac{d}{dx}(\sin(\sqrt{x})) = \cos(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) \]

### Step 4: Differentiate \( \sqrt{x} \)
Now, we differentiate \( \sqrt{x} \):
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]

### Step 5: Combine the results
Now we can substitute back into our derivative:
\[ \frac{d}{dx}(\sin(\sqrt{x})) = \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]
Substituting this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = e^{\sin(\sqrt{x})} \cdot \left(\cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}\right) \]

### Final Result
Thus, we have:
\[ \frac{dy}{dx} = \frac{e^{\sin(\sqrt{x})} \cdot \cos(\sqrt{x})}{2\sqrt{x}} \]

### Summary of the Steps
1. Differentiate the outer function \( e^{u} \).
2. Differentiate the inner function \( \sin(\sqrt{x}) \) using the chain rule.
3. Differentiate \( \sqrt{x} \).
4. Combine the results to get the final derivative.