Differentiate the following function w.r.t. x:
`x^(2) cos 5x`

Answer

To differentiate the function \( y = x^2 \cos(5x) \) with respect to \( x \), 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:

\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step-by-Step Solution:

1. **Identify the Functions**:
Let \( u = x^2 \) and \( v = \cos(5x) \).

2. **Differentiate \( u \)**:
We need to find \( \frac{du}{dx} \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^2) = 2x \]

3. **Differentiate \( v \)**:
We need to find \( \frac{dv}{dx} \):
\[ \frac{dv}{dx} = \frac{d}{dx}(\cos(5x)) = -\sin(5x) \cdot \frac{d}{dx}(5x) = -5\sin(5x) \]

4. **Apply the Product Rule**:
Now, we apply the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substituting the values we found:
\[ \frac{dy}{dx} = x^2 \cdot (-5\sin(5x)) + \cos(5x) \cdot (2x) \]

5. **Simplify the Expression**:
Combine the terms:
\[ \frac{dy}{dx} = -5x^2 \sin(5x) + 2x \cos(5x) \]

Thus, the derivative of the function \( y = x^2 \cos(5x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 2x \cos(5x) - 5x^2 \sin(5x) \]