1. ## logarithmic differentiation

why do we have to rewrite as $|y|=|x|^2|3x-1|^3$ ?

2. ## Re: logarithmic differentiation

Because log function is not defined for negative numbers

3. ## Re: logarithmic differentiation

isnt it that $ln|x|=\frac{1}{x}$ both $lnx$ and $ln(-x)$ equals to $\frac{1}{x}$, so it doesn't matter?
and also, when it comes to rewriting using modulus, how do i know for sure the correct way to write it? why not $|y|=|x^5(3x-1)^3|$
or $|y|=|x|^5(3|x|-1)^3$

5. ## Re: logarithmic differentiation

oh yeah thanks and sorry it should be $(ln|x|)'=\frac{1}{x}$

6. ## Re: logarithmic differentiation

No, it should be $(ln|x|)'= \frac{1}{|x|}$.

7. ## Re: logarithmic differentiation

Originally Posted by HallsofIvy
No, it should be $(ln|x|)'= \frac{1}{|x|}$.
why do you say so?

8. ## Re: logarithmic differentiation

i mean, is there some significance?? it seems the same to me in the end

9. ## Re: logarithmic differentiation

Draw the graph of ln(x). It's derivative is always increasing so 1/x has to be positive for the domain of the original function. You can also draw the graph of 1/x. There is a positive and negative portion of the graph. The absolute value keeps only the positive portion. The first derivative tells you how the original function changes or how it's slope changes. As you get close to zero the slope gets really steep. Try putting really small positive numbers into 1/x. What happens?

10. ## Re: logarithmic differentiation

Originally Posted by wondering
Draw the graph of ln(x). It's derivative is always increasing so 1/x has to be positive for the domain of the original function. You can also draw the graph of 1/x. There is a positive and negative portion of the graph. The absolute value keeps only the positive portion. The first derivative tells you how the original function changes or how it's slope changes. As you get close to zero the slope gets really steep. Try putting really small positive numbers into 1/x. What happens?
?? u get a huge number approaching infinity? but how is that related to what we're discussing?

11. ## Re: logarithmic differentiation

Originally Posted by muddywaters
it should be $(\ln|x|)'=\frac{1}{x}$
That is correct. See here.

12. ## Re: logarithmic differentiation

You get a really small number approaching infinity and a really big number as you approach zero. Does that match how the slope of ln(x) changes from (0,inf)?