Question

Find dy/dx if y = ln[(square root of x)/(5 - x)]

Answer 1

\[y=\frac{\ln \sqrt{x}}{5-x}=\frac{\frac{1}{2}\ln x}{5-x}=\frac{\ln x}{10-2x}\]
use this rule
\[(\frac{f}{g})'=\frac{f'g-fg'}{g^2}\]

Answer 2

@Jonask, it looks like the log contains all of \(\dfrac{\sqrt x}{5-x}\).

Answer 3

I would make use of log properties:
\[y=\ln\left[\frac{\sqrt x}{5-x}\right]\\
~~=\frac{1}{2}\ln x-\ln(5-x)\]

Answer 4

ohhh sry

Answer 5

but your right thats the way we shud do it

Answer 6

Quotient rule can get messy pretty quickly :)

Answer 7

now use
\[\frac{ d }{ dx }\ln f(x)=\frac{ f'(x) }{ f(x) }\]

Answer 8

from here
\[\frac{ 1 }{ 2 }\ln x-\ln (5-x)\]

Answer 9

The second one would be 1/(5-x), right? I'm just not sure what to do with the first one because of its 1/2. Would it just be 1/2 * (1/x)? Or am I doing it entirely wrong? xD

Answer 10

yes your right you do it as if the 1/2 is not there 1/2*(1/x) is correct

Answer 11

Well, I can't find anyway of simplifying it without making a huge mess, so do ya think y = (1/2x) - (1/5 - x) would be an acceptable final answer?

Answer 12

yes \[\frac{ 1 }{ 2x }-\frac{ 1 }{ 5-x }=\frac{ 5+x }{ 10x-2x^2}\]

Answer 13

Alright. Thank you very much! :D

Answer 14

i think ur missing a "-" sign there derivative of ln(5-x)=-1/(5-x)

Answer 15

yes