If `y=x^(3)cosx" then "(dy)/(dx)`=……………………

Options

A. `x^(2)(3cosx-xsinx)`
B. `x^(2)(3cosx+xsinx)`
C. `3x^(2).cosx+x^(3)sinx`
D. None of these

Answer

To find the derivative of the function \( y = x^3 \cos x \), we will use the product rule of differentiation. The product rule states that if you have two functions multiplied together, \( u \) and \( v \), then the derivative \( \frac{dy}{dx} \) 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 \( u = x^3 \) and \( v = \cos x \).

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) is:
\[ u' = \frac{d}{dx}(x^3) = 3x^2 \]
- The derivative of \( v \) is:
\[ v' = \frac{d}{dx}(\cos x) = -\sin x \]

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

4. **Simplify the Expression**:
- Combine the terms:
\[ \frac{dy}{dx} = 3x^2 \cos x - x^3 \sin x \]

5. **Factor Out Common Terms**:
- Notice that \( x^2 \) is a common factor:
\[ \frac{dy}{dx} = x^2(3 \cos x - x \sin x) \]

### Final Answer:
\[ \frac{dy}{dx} = x^2(3 \cos x - x \sin x) \]