Question

Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72.

$f\left(x\right)=\left|x\right|$

Short answer

The derivative of the function is$f\text{'}\left(x\right)=\left\{\begin{array}{l}-1x<0\\ DNEx=0\\ 1x>0\end{array}\right.$.

Step-by-step solution

Step 1. Given Information

The given function is$f\left(x\right)=\left|x\right|$.

Step 2. Rewrite the function

The given function can be rewritten as,

$f\left(x\right)=\left\{\begin{array}{l}-xx\le 0\\ xx>0\end{array}\right.$

Step 3. Find the derivative

• It is known that, the derivative of the identity function is $\frac{d}{dx}\left(x\right)=1$.
• Find the derivative for localid="1648725381209" $x<0$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}(-x)\\ & =& -1\end{array}$

• Find the derivative for $x>0$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}\left(x\right)\\ & =& 1\end{array}$

• Since $-1\ne 1$, the derivatives at left and right pieces are not equal at $x=0$. The derivative does not exists at $x=0$.
• So, the derivative of the function is,$f\text{'}\left(x\right)=\left\{\begin{array}{l}-1x<0\\ DNEx=0\\ 1x>0\end{array}\right.$.