Using the quotient rule to find $f'$ where $f(x) = \dfrac{6x}{5 \ln(x)}$ Below is a problem and the process by which the first derivative is found:



I used the quotient rule but did not get the same result. Can someone explain this to me?

Here is what I got after using the quotient rule:
Your answer is the same as your professor's. $\frac{30\ln{x} - 30}{(5\ln{x})^2} = \frac{30\ln{x} - 30}{25(\ln{x})^2} = \frac{6\ln{x} - 6}{5(\ln{x})^2}$. Splitting the numerator and simplifying slightly, we get $\frac{6}{5(\ln{x})} - \frac{6}{5(\ln{x})^2} = \frac{6}{5}\left((\ln{x})^{-1} - (\ln{x})^{-2}\right)$.

Just because answers look different doesn't mean they are. If your answer looks different from someone else's, you should compare your steps against theirs, not assume that one answer is right and the other is wrong.
$$
  f(x) = \frac{6 x}{5 \ln (x)}
$$

$$
 f'(x) = \frac{6}{5} \left( \frac{1}{\ln (x)}-\frac{1}{ \ln ^2(x)} \right)
$$

$$
f''(x) = \frac{6}{5} \left( \frac{2}{x\ln^{3} (x)}-\frac{1}{ x\ln ^2(x)} \right)
$$