Differentiate the following w.r.t. x :
`e^(sinsqrtx)`

Answer

To differentiate the function \( y = e^{\sin(\sqrt{x})} \) with respect to \( x \), we will use the chain rule. Here are the steps:

### Step 1: Identify the outer and inner functions
Let \( y = e^{u} \) where \( u = \sin(\sqrt{x}) \).

### Step 2: Differentiate the outer function
The derivative of \( e^{u} \) with respect to \( u \) is \( e^{u} \). Thus, we have:
\[ \frac{dy}{du} = e^{\sin(\sqrt{x})} \]

### Step 3: Differentiate the inner function
Next, we differentiate \( u = \sin(\sqrt{x}) \) with respect to \( x \). We will again use the chain rule:
- The derivative of \( \sin(v) \) is \( \cos(v) \) where \( v = \sqrt{x} \).
- The derivative of \( v = \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \).

Thus, we have:
\[ \frac{du}{dx} = \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

### Step 4: Apply the chain rule
Now, we apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = e^{\sin(\sqrt{x})} \cdot \left( \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \right) \]

### Step 5: Simplify the expression
Putting it all together, we get:
\[ \frac{dy}{dx} = \frac{e^{\sin(\sqrt{x})} \cos(\sqrt{x})}{2\sqrt{x}} \]

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

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