If `int f(x) dx = 2 cos sqrt(x) + c`, then f(x) =

Options

A. `sin sqrt(x)`
B. `-(sin sqrt(x))/(sqrt(x))`
C. `2 cos sqrt(x)`
D. None of these

Answer

To find \( f(x) \) given that

\[ \int f(x) \, dx = 2 \cos(\sqrt{x}) + c, \]

we need to differentiate the right-hand side with respect to \( x \).

### Step 1: Differentiate the given integral
We start by differentiating both sides:

\[ f(x) = \frac{d}{dx} \left( 2 \cos(\sqrt{x}) + c \right). \]

### Step 2: Apply the chain rule
Using the chain rule for differentiation, we differentiate \( 2 \cos(\sqrt{x}) \):

\[ f(x) = 2 \cdot \frac{d}{dx} \left( \cos(\sqrt{x}) \right). \]

Now, we apply the chain rule:

\[ \frac{d}{dx} \left( \cos(\sqrt{x}) \right) = -\sin(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}). \]

### Step 3: Differentiate \( \sqrt{x} \)
The derivative of \( \sqrt{x} \) is:

\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}. \]

### Step 4: Substitute back into the equation
Now substituting this back into our expression for \( f(x) \):

\[ f(x) = 2 \cdot \left( -\sin(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \right). \]

### Step 5: Simplify the expression
This simplifies to:

\[ f(x) = -\frac{\sin(\sqrt{x})}{\sqrt{x}}. \]

### Final Result
Thus, we have:

\[ f(x) = -\frac{\sin(\sqrt{x})}{\sqrt{x}}. \]

### Conclusion
Now, we can check the options provided. The correct option matches with:

\[ \text{Option 2: } -\frac{\sin(\sqrt{x})}{\sqrt{x}}. \]