Differentiate `x^(2)cos x.`

Options

A. `2xcos x-x^(1)sinx.`
B. `xcos x-x^(2)sinx.`
C. `2xcos x-x^(2)sinx.`
D. `2x sinx-x^(2)cosx.`

Explanation

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

\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step-by-Step Solution:

1. **Identify the functions**:
Let \( u = x^2 \) and \( v = \cos x \).

2. **Differentiate \( u \)**:
The derivative of \( u \) with respect to \( x \) is:
\[ \frac{du}{dx} = 2x \]

3. **Differentiate \( v \)**:
The derivative of \( v \) with respect to \( x \) is:
\[ \frac{dv}{dx} = -\sin x \]

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

5. **Simplify the expression**:
This simplifies to:
\[ \frac{dy}{dx} = -x^2 \sin x + 2x \cos x \]

6. **Final result**:
Therefore, the derivative of \( y = x^2 \cos x \) is:
\[ \frac{dy}{dx} = 2x \cos x - x^2 \sin x \]