Differentiate the following functions w.r.t. x.
`(2x +3) (x^3 - x+ 2)`

Answer

To differentiate the function \( (2x + 3)(x^3 - x + 2) \) with respect to \( x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

\[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \]

### Step-by-Step Solution:

1. **Identify the Functions**:
Let \( u = 2x + 3 \) and \( v = x^3 - x + 2 \).

2. **Differentiate \( u \) and \( v \)**:
- Differentiate \( u \):
\[ u' = \frac{d}{dx}(2x + 3) = 2 \]
- Differentiate \( v \):
\[ v' = \frac{d}{dx}(x^3 - x + 2) = 3x^2 - 1 \]

3. **Apply the Product Rule**:
Now, apply the product rule:
\[ \frac{d}{dx}[(2x + 3)(x^3 - x + 2)] = u'v + uv' \]
Substituting the values we found:
\[ = (2)(x^3 - x + 2) + (2x + 3)(3x^2 - 1) \]

4. **Expand the Expression**:
- First term:
\[ 2(x^3 - x + 2) = 2x^3 - 2x + 4 \]
- Second term:
\[ (2x + 3)(3x^2 - 1) = 2x(3x^2) + 2x(-1) + 3(3x^2) + 3(-1) = 6x^3 - 2x + 9x^2 - 3 \]

5. **Combine Like Terms**:
Now combine all the terms:
\[ 2x^3 - 2x + 4 + 6x^3 - 2x + 9x^2 - 3 \]
Combine like terms:
\[ (2x^3 + 6x^3) + 9x^2 + (-2x - 2x) + (4 - 3) = 8x^3 + 9x^2 - 4x + 1 \]

6. **Final Answer**:
Therefore, the derivative of the function \( (2x + 3)(x^3 - x + 2) \) is:
\[ \frac{d}{dx}[(2x + 3)(x^3 - x + 2)] = 8x^3 + 9x^2 - 4x + 1 \]