Question

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

$f\left(x\right)=|3x+1|$

Short answer

The derivative of the function is, $f\text{'}\left(x\right)=\left\{\begin{array}{l}-3,ifx<-\frac{1}{3}\\ DNE,ifx=-\frac{1}{3}\\ 3,ifx>-\frac{1}{3}\end{array}\right.$.

Step-by-step solution

Step 1. Given Information

The given function is$f\left(x\right)=|3x+1|$.

Step 2. Rewrite the function

The given function can be rewritten as,$f\left(x\right)=\left\{\begin{array}{l}-(3x+1),ifx\le \frac{-1}{3}\\ 3x+1,ifx>\frac{-1}{3}\end{array}\right.$.

Step 3. Find the derivative

• It is known that, the derivative of the linear function is $(mx+b)\text{'}=m$.
• Find the derivative for $x<\frac{-1}{3}$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}(-(3x+1\left)\right)\\ & =& \frac{d}{dx}(-3x-1)\\ & =& -3\end{array}$

• Find the derivative for $x>\frac{-1}{3}$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}(3x+1)\\ & =& 3\end{array}$

• Since $-3\ne 3$, the derivatives at left and right pieces are not equal at $x=-\frac{1}{3}$. The derivative does not exists at $x=-\frac{1}{3}$.
• So, the derivative of the function is, $f\text{'}\left(x\right)=\left\{\begin{array}{l}-3,ifx<-\frac{1}{3}\\ DNE,ifx=-\frac{1}{3}\\ 3,ifx>-\frac{1}{3}\end{array}\right.$.