Question

Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.

$f\left(x\right)=\left\{\ln \right(-x),\mathrm{if}x<0;\ln x,\mathrm{if}x\ge 0\}$

Short answer

The derivative of function is

$$\begin{gathered} f\left(x\right)=\frac{1}{x},x<0 \\ DNE,x=0 \\ \frac{1}{x},x>0 \end{gathered}$$

Step-by-step solution

Step 1. Given Information 

The given function is $f\left(x\right)=\left\{\ln \right(-x),\mathrm{if}x<0;\ln x,\mathrm{if}x\ge 0\}$

Step 2. Calculation 

From the differentiation rules,

$$\begin{gathered} \frac{d}{dx}\left(\ln \right(-x\left)\right)=\frac{1}{x} \\ \frac{d}{dx}\left(\ln x\right)=\frac{1}{x} \\ \mathrm{Let}g\left(x\right)=\ln (-x),h\left(x\right)=\ln \left(x\right) \\ \mathrm{And}g\text{'}\left(x\right)=\frac{1}{x},h\text{'}\left(x\right)=\frac{1}{x} \\ \mathrm{When}x=0\mathrm{then}, \\ g\text{'}\left(0\right)=h\text{'}\left(0\right)=\mathrm{undefined} \end{gathered}$$

It follows that the function is not continuous at $x=0$.

Thus, the derivative of function is,

$$\begin{gathered} f\left(x\right)=\frac{1}{x},x<0 \\ DNE,x=0 \\ \frac{1}{x},x>0 \end{gathered}$$