# Math Help - Derivative question Help

1. ## Derivative question Help

I got my recent test back and one of the questions was to find dy/dx of the following:
ln( )... Sorry i had trouble writing it out formaly but its ln of whats in the bracket.

I seemed to solved it but i just ended up with a HUGE long denominator and ended up getting 2 marks out of 5 soo i was wondering if someone would be kind enough to solve this with the steps soo i can see where i went wrong. There are soo many brackets im just lost trying to trace it back. Thanks would be appriciated

2. I'm assuming you mean to take the derivative of

$\ln\left[\dfrac{e^{2x}\sin(x)}{x^{3}\sqrt{x^{2}+1}}\right],$ correct? If so, you need to use the chain rule first, then quotient, etc. What steps do you get?

3. What I would do is write

$\ln\left(\dfrac{e^{2x}\sin(x)}{x^{3}\sqrt{x^{2}+1} }\right)=\ln e^{2x}+\ln \sin x-\ln x^3 -\dfrac{1}{2}\ln (x^2+1)=2x+\ln \sin x-3\ln x-\dfrac{1}{2}\ln (x^2+1)$

4. Originally Posted by Ackbeet
I'm assuming you mean to take the derivative of

$\ln\left[\dfrac{e^{2x}\sin(x)}{x^{3}\sqrt{x^{2}+1}}\right],$ correct? If so, you need to use the chain rule first, then quotient, etc. What steps do you get?
Yes correct, im new with this site soo i dont no how to formaly write it and the site i was using to change it made my ln into log
umm i basically did ln(u) and then solved the inside (u), seperatly and then subed it back in ln so i did 1/(my answer of dx/dy of u).
Got a huge mess because of the quotient rule and chain rules.

Im actually surprised i never thought of writing it the way Fobos3 wrote it.. seems like a correct way and much easier

5. fobos3's method is certainly valid as well. Warning: sometimes fobos3's method winds up being just as much work, because in order to simplify your answer you have to re-combine things. In this case, though, it does appear his method would be easier. Take your pick.

6. Originally Posted by Ackbeet
fobos3's method is certainly valid as well. Warning: sometimes fobos3's method winds up being just as much work, because in order to simplify your answer you have to re-combine things. In this case, though, it does appear his method would be easier. Take your pick.
I just have one more quick question... suppose the terms inside are added together, you can no longer use fobos3's method correct? Only ln(xy)=lnx+lny.. so what happens if we had ln(x+y) can you split to to lnx+lny?
For example : ln(x^2+x+1).. soo you cant write lnx^2+lnx+ln1 right?

7. The logarithm turns multiplication into addition, not the reverse. So your hunch is correct: fobos3's method will not work if the argument of the logarithm is a bunch of terms added together. You'll have to use the chain rule.