Differentiate `(x^2-5x+8)(x^3+7x+9)` in three ways mentioned below:
(i) by using product rule
(ii) by expanding the product to obtain a single polynomial.
(iii) by logarithmic differentiation.
Do they all give the same answer?

Explanation

To differentiate the function \( f(x) = (x^2 - 5x + 8)(x^3 + 7x + 9) \) using three different methods, we will follow the steps outlined below:

### (i) Differentiation using the Product Rule

1. **Identify the functions**: Let \( u = x^2 - 5x + 8 \) and \( v = x^3 + 7x + 9 \).
2. **Differentiate \( u \) and \( v \)**:
- \( \frac{du}{dx} = 2x - 5 \)
- \( \frac{dv}{dx} = 3x^2 + 7 \)
...