Question

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

$f\left(x\right)=\left|1-2x\right|$

Short answer

The derivative of the function is$f\text{'}\left(x\right)=\left\{\begin{array}{l}2,ifx>\frac{1}{2}\\ DNE,ifx=\frac{1}{2}\\ -2.ifx<\frac{1}{2}\end{array}\right.$.

Step-by-step solution

Step 1. Given Information

The given function is$f\left(x\right)=\left|1-2x\right|$.

Step 2. Rewrite the function

The given function can be rewritten as, $f\left(x\right)=\left\{\begin{array}{l}-(1-2x),ifx\ge \frac{1}{2}\\ 1-2x,ifx>\frac{1}{2}\end{array}\right.$.

Step 3. Find the derivative

• It is known that, the derivative of the linear function is $(mx+b)\text{'}=m$.
• Find the derivative for $x>\frac{1}{2}$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}(-(1-2x\left)\right)\\ & =& \frac{d}{dx}(2x-1)\\ & =& 2\end{array}$

• Find the derivative for $x>\frac{1}{2}$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}(1-2x)\\ & =& -2\end{array}$

• Since $2\ne -2$, the derivatives at left and right pieces are not equal at $x=\frac{1}{2}$. The derivative does not exists at $x=\frac{1}{2}$.
• So, the derivative of the function is, $f\text{'}\left(x\right)=\left\{\begin{array}{l}2,ifx>\frac{1}{2}\\ DNE,ifx=\frac{1}{2}\\ -2.ifx<\frac{1}{2}\end{array}\right.$.