One problem I've been asked to solve is giving me some trouble on the particular sequence to solve. Find the differential of
$$g(t)=\frac{10 \log_{4}t}{t}$$
Looking at the problem, you can see that the 10 is a constant, and can be pulled out of the function and that the quotient rule applies
$$g'(t)= 10[\frac{\log_{4}t}{t}]$$ $$g'(t)=10[\frac{\log_{4}t[\frac{d}{dx}(t)]-t[\frac{d}{dx}\log_4t]}{t^2}]$$ $$g'(t)= 10[\frac{\log_4t-t(\frac{1}{\ln(4)t})}{t^2}]$$ $$g'(t)= 10[\frac{\log_4t-\frac{t}{\ln(4)t}}{t^2}]$$ $$g'(t)=10[\frac{t^2}{\log_4- 2t \ln(4)}]$$
I got stuck right there, since I'm not sure my last line works, so I took a look at the book solution, which is shown below:
$$g(t)= \frac{10 \log_4t}{t} = \frac{10}{\ln 4}\cdot\frac{\ln(t)}{t}$$
So my questions are:
$\cdot$There are 3 elements of the numerator, so how does the $\log_4$ morph into the denominator of $\frac{10}{\ln 4}$?
$\cdot$The book example shows that the quotient differentiation is done only on $\frac{\ln(t)}{t}$, but why not on the other fraction? Is it due to those values being constants and not open to differentiation?
$\cdot$Is the last line of my own work accurate?
