Question

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

$f\left(x\right)=\left\{\begin{array}{l}{x}^3,ifx<1\\ x,ifx\ge 1\end{array}\right.$

Short answer

The derivative of the function is$f\text{'}\left(x\right)=\left\{\begin{array}{l}3x^2,ifx<1\\ DNE,ifx=1\\ 1,ifx>1\end{array}\right.$.

Step-by-step solution

Step 1. Given Information

The given function is$f\left(x\right)=\left\{\begin{array}{l}{x}^3,ifx<1\\ x,ifx\ge 1\end{array}\right.$.

Step 2. Find the derivative

• It is known that, the power rule of derivative is $\left(x^n\right)\text{'}=n{x}^{n-1}$.
• Find the derivative for $x<1$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}\left(x^3\right)\\ & =& 3x^2\end{array}$

• Find the derivative for $x>1$.

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

• Since $3{\left(1\right)}^2\ne 1$, the derivatives at left and right pieces are not equal at $x=1$. The derivative does not exists at $x=1$.
• So, the derivative of the function is, $f\text{'}\left(x\right)=\left\{\begin{array}{l}3x^2,ifx<1\\ DNE,ifx=1\\ 1,ifx>1\end{array}\right.$.