If $f(x) = \left(x + 1\right)^{2} \left(x + 2\right)^{3}$, what is $f'\left(0\right)$?

Explanation

1. 01

   $\log y = 2\log(x + 1) + 3\log(x + 2)$, so $\frac{y'}{y} = \frac{2}{x + 1} + \frac{3}{x + 2}$.

2. 02

   At 0: $y = 8$ and $\frac{y'}{y} = 2 + \frac{3}{2} = \frac{7}{2}$, so $y' = 28$.