Find the derivatives of the following :
`(x^(2)+2)cosx`

Answer

To find the derivative of the function \( y = (x^2 + 2) \cos x \), we will use the product rule of differentiation. The product rule states that if you have a function that is the product of two functions \( u \) and \( v \), then the derivative is given by:

\[ \frac{dy}{dx} = u'v + uv' \]

where \( u' \) is the derivative of \( u \) and \( v' \) is the derivative of \( v \).

### Step-by-step Solution:

1. **Identify the functions:**
Let
\[ u = x^2 + 2 \quad \text{and} \quad v = \cos x \]

2. **Find the derivatives of \( u \) and \( v \):**
- The derivative of \( u \) is:
\[ u' = \frac{d}{dx}(x^2 + 2) = 2x \]
- The derivative of \( v \) is:
\[ v' = \frac{d}{dx}(\cos x) = -\sin x \]

3. **Apply the product rule:**
Now, we can apply the product rule:
\[ \frac{dy}{dx} = u'v + uv' \]
Substituting the values we found:
\[ \frac{dy}{dx} = (2x)(\cos x) + (x^2 + 2)(-\sin x) \]

4. **Simplify the expression:**
\[ \frac{dy}{dx} = 2x \cos x - (x^2 + 2) \sin x \]

5. **Final expression:**
Thus, the derivative of the function \( y = (x^2 + 2) \cos x \) is:
\[ \frac{dy}{dx} = 2x \cos x - (x^2 + 2) \sin x \]