If $f(x) = \sqrt{\frac{(x + 1)(x + 9)}{x + 4}}$, what is $f'\left(0\right)$?

Explanation

1. 01

   $\log y = \frac{1}{2}\left[\log(x + 1) + \log(x + 9) - \log(x + 4)\right]$, so at 0, $\frac{y'}{y} = \frac{1}{2}\left(1 + \frac{1}{9} - \frac{1}{4}\right) = \frac{31}{72}$.

2. 02

   $y(0) = \sqrt{\frac{9}{4}} = \frac{3}{2}$, so $f'(0) = \frac{3}{2} \times \frac{31}{72} = \frac{31}{48}$.