Does this statement about natural logs of functions make sense?

Question

For two functions $a(x)$ and $b(x)$, since $\ln(a/b) = \ln(a) - \ln(b)$, does $$\frac{d}{dx} \ln(a/b) = \frac{d}{dx} \ln(a) - \frac{d}{dx} \ln(b) = \frac{1}{a} \cdot \frac{da}{dx} - \frac{1}{b} \cdot \frac{db}{dx}?$$

Answer 1

Looks good to me.

It is probably clearer from a logical perspective if you wrote the whole thing with $a(x)$ and $b(x)$ instead of $a$ and $b$, except for the derivatives $\frac{da}{dx}$ and $\frac{db}{dx}$ of course. That way, every step can be justified by an appropriate law of algebra or calculus.

Answer 2

Makes perfect sense.

Even if you reverse the order of differentiation you get the the very same result. First differentiate log and next apply quotient rule.When primes are with respect to x,and omitting argument we have

$$ \dfrac {b}{a}\cdot \dfrac {ba^{'}-b^{'}a}{b^2}$$ $$=\dfrac{a^{'}}{a}- \dfrac{b^{'}}{b} $$