Differentiating an absolute value...

Oct 2012
44
0
United States
I love math, I really do, but differentiating absolute values is driving me crazy! I just don't get it, and the book does a terrible job explaining it. I have a test tomorrow which is guaranteed to have an absolute value differentiation question on it. :( :( :(

I would truly appreciate it if someone could walk me through this example:

Differentiate: y = |2x - 3|
 

Prove It

MHF Helper
Aug 2008
12,897
5,001
I love math, I really do, but differentiating absolute values is driving me crazy! I just don't get it, and the book does a terrible job explaining it. I have a test tomorrow which is guaranteed to have an absolute value differentiation question on it. :( :( :(

I would truly appreciate it if someone could walk me through this example:

Differentiate: y = |2x - 3|
First of all, \(\displaystyle \displaystyle \begin{align*} |X| = \begin{cases} \phantom{-}X \textrm{ if } X \geq 0 \\ -X \textrm{ if } X < 0 \end{cases} \end{align*}\) and also note that \(\displaystyle \displaystyle \begin{align*} |X| \end{align*}\) is NOT differentiable at \(\displaystyle \displaystyle \begin{align*} X = 0 \end{align*}\).

So for your function

\(\displaystyle \displaystyle \begin{align*} y &= \left| 2x - 3 \right| \\ &= \begin{cases} 2x - 3 \textrm{ if } 2x - 3 \geq 0 \\ -\left( 2x - 3 \right) \textrm{ if } 2x - 3 < 0 \end{cases} \\ &= \begin{cases} 2x - 3 \textrm{ if } x \geq \frac{3}{2} \\ 3 - 2x \textrm{ if } x < \frac{3}{2} \end{cases} \end{align*}\)

So differentiate each piece, and remember that the function won't be differentiable at \(\displaystyle \displaystyle \begin{align*} x = \frac{3}{2} \end{align*}\).