The formula under discussion is this:
$$ \frac{d}{dx} [ \log_b x] = \frac {1}{x \ln b}$$
The author does not derive the formula completely, instead he states that:
$$\frac {d}{dx} [\log_b x ] = \frac {d}{dx} \left[\frac{\ln x}{\ln b}\right] \qquad\qquad (1)$$
I understand it up to this. You let $\log_b x$ equal to $y$ and then $y$ can be rewritten as $b^y$ and it equals $x.$ After that simply taking the natural log on both sides will arrange it into the form expressed in (1).
However after this the author states:
$$ \frac {d}{dx} \left[\frac{\ln x}{\ln b}\right] = \frac{1}{\ln b} \frac{d}{dx} [\ln x]$$
This is the step that I don't understand. I would suppose that the quotient rule would be applied but that just makes it all very weird and I can't get the L.H.S.
I would appreciate if anyone can explain what is exactly happening here.