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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?

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    About your various questions, there are some errors, including algebra errors. You quoted the derivative of $\log_4 t$ correctly, presumably from a list of formulas. I would not be able to remember such a thing, so I would always derive it as in the comment above.2011-07-17

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The expression $\log_4(t)$ is not multiplication. It is the application of a function. Your mystery will be revealed by the change of base formula, $\log_4(t) = {\log(t)\over \log(4)}.$