# Question #f3133

Jul 23, 2016

$= \frac{2}{x}$

#### Explanation:

$\frac{d}{\mathrm{dx}} \left(\ln f \left(x\right)\right) = \frac{1}{f \left(x\right)} f ' \left(x\right)$

So here that means

$\frac{d}{\mathrm{dx}} \left(\ln 3 {x}^{2}\right) q \quad \star$

$= \frac{1}{3 {x}^{2}} \cdot 6 x$

$= \frac{2}{x}$

Just to play about a bit, we can re-write $\star$ as

$\frac{d}{\mathrm{dx}} \left(2 \ln \sqrt{3} x\right)$

$= 2 \cdot \frac{1}{\sqrt{3} x} \cdot \sqrt{3}$

$= \frac{2}{x}$

or we can rewrite $\star$ as

$\frac{d}{\mathrm{dx}} \left(\ln 3 + \ln {x}^{2}\right)$

$= \frac{d}{\mathrm{dx}} \left(\ln 3 + 2 \ln x\right)$

$= 0 + 2 \cdot \frac{1}{x} = \frac{2}{x}$