Q. 65 - Calculus

Question

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

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

Step-by-Step Solution

Answer

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

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, 

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

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 $x<0$.

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

• 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, $f\text{'}\left(x\right)=\left\{\begin{array}{l}-1 x<0\\ DNE x=0\\ 1 x>0\end{array}\right.$.