`f(x)=cossqrtx`. find `dy/dx`

Answer

To find the derivative of the function \( f(x) = \cos(\sqrt{x}) \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
In the function \( f(x) = \cos(\sqrt{x}) \):
- The outer function is \( \cos(u) \) where \( u = \sqrt{x} \).
- The inner function is \( u = \sqrt{x} \).

### Step 2: Differentiate the outer function
The derivative of the outer function \( \cos(u) \) with respect to \( u \) is:
\[ \frac{d}{du}[\cos(u)] = -\sin(u) \]

### Step 3: Differentiate the inner function
Next, we differentiate the inner function \( u = \sqrt{x} \):
\[ \frac{d}{dx}[\sqrt{x}] = \frac{1}{2\sqrt{x}} \]

### Step 4: Apply the chain rule
Using the chain rule, we combine the derivatives of the outer and inner functions:
\[ \frac{dy}{dx} = \frac{d}{du}[\cos(u)] \cdot \frac{du}{dx} = -\sin(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

### Step 5: Write the final answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{\sin(\sqrt{x})}{2\sqrt{x}} \]

### Summary of the solution:
The final result is:
\[ \frac{dy}{dx} = -\frac{\sin(\sqrt{x})}{2\sqrt{x}} \]