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

Printable View

- May 25th 2013, 07:40 PMmuddywaterslogarithmic differentiation
why do we have to rewrite as $\displaystyle |y|=|x|^2|3x-1|^3 $ ?

- May 25th 2013, 09:01 PMibduttRe: logarithmic differentiation
Because log function is not defined for negative numbers

- May 25th 2013, 10:56 PMmuddywatersRe: logarithmic differentiation
isnt it that $\displaystyle ln|x|=\frac{1}{x}$ both $\displaystyle lnx$ and $\displaystyle ln(-x)$ equals to $\displaystyle \frac{1}{x}$, 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 $\displaystyle |y|=|x^5(3x-1)^3|$

or $\displaystyle |y|=|x|^5(3|x|-1)^3$ - May 26th 2013, 03:04 AMibduttRe: logarithmic differentiation
- May 26th 2013, 04:29 AMmuddywatersRe: logarithmic differentiation
oh yeah thanks and sorry it should be $\displaystyle (ln|x|)'=\frac{1}{x} $

- May 26th 2013, 05:53 AMHallsofIvyRe: logarithmic differentiation
No, it should be $\displaystyle (ln|x|)'= \frac{1}{|x|}$.

- May 26th 2013, 06:33 AMmuddywatersRe: logarithmic differentiation
- May 26th 2013, 06:35 AMmuddywatersRe: logarithmic differentiation
i mean, is there some significance?? it seems the same to me in the end

- May 26th 2013, 06:47 AMwonderingRe: 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?

- May 26th 2013, 07:24 AMmuddywatersRe: logarithmic differentiation
- May 26th 2013, 07:52 AMPlatoRe: logarithmic differentiation
**That is correct**. See here. - May 26th 2013, 07:58 AMwonderingRe: 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)?