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 these steps:

### (i) Differentiating using the Product Rule

1. **Identify the functions**:
Let \( u = x^2 - 5x + 8 \) and \( v = x^3 + 7x + 9 \).

2. **Find the derivatives**:
...