Find the derivatives of the following at any point of their domains :
`f(x)=(x^(2)-5x+6)(x^(3)+2)`

Answer

To find the derivative of the function \( f(x) = (x^2 - 5x + 6)(x^3 + 2) \), 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:

\[ (uv)' = u'v + uv' \]

### Step-by-step solution:

1. **Identify the functions**:
Let \( u = x^2 - 5x + 6 \) and \( v = x^3 + 2 \).

2. **Differentiate \( u \)**:
\[ u' = \frac{d}{dx}(x^2 - 5x + 6) = 2x - 5 \]

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

4. **Apply the product rule**:
Using the product rule, we have:
\[ f'(x) = u'v + uv' \]

5. **Substitute \( u, u', v, v' \)**:
\[ f'(x) = (2x - 5)(x^3 + 2) + (x^2 - 5x + 6)(3x^2) \]

6. **Expand the first term**:
\[ (2x - 5)(x^3 + 2) = 2x \cdot x^3 + 2 \cdot 2 - 5 \cdot x^3 - 5 \cdot 2 = 2x^4 + 4 - 5x^3 - 10 = 2x^4 - 5x^3 - 6 \]

7. **Expand the second term**:
\[ (x^2 - 5x + 6)(3x^2) = 3x^2 \cdot x^2 - 5 \cdot 3x^2 \cdot x + 6 \cdot 3x^2 = 3x^4 - 15x^3 + 18x^2 \]

8. **Combine the two expanded terms**:
\[ f'(x) = (2x^4 - 5x^3 - 6) + (3x^4 - 15x^3 + 18x^2) \]

9. **Combine like terms**:
\[ f'(x) = (2x^4 + 3x^4) + (-5x^3 - 15x^3) + 18x^2 - 6 = 5x^4 - 20x^3 + 18x^2 - 6 \]

### Final Answer:
The derivative of the function \( f(x) = (x^2 - 5x + 6)(x^3 + 2) \) is:
\[ f'(x) = 5x^4 - 20x^3 + 18x^2 - 6 \]