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|**

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 $\displaystyle \displaystyle \begin{align*} x = \frac{3}{2} \end{align*}$.

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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?