# Differentiating an absolute value...

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• Oct 29th 2012, 05:38 PM
theunforgiven
Differentiating an absolute value...
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|
• Oct 29th 2012, 06:04 PM
Prove It
Re: Differentiating an absolute value...
Quote:

Originally Posted by theunforgiven
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*}.
• Oct 29th 2012, 07:17 PM
Plato
Re: Differentiating an absolute value...
Quote:

Originally Posted by theunforgiven
I would truly appreciate it if someone could walk me through this example:
Differentiate: y = |2x - 3|

Graph this.

What are the slops?