Question

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

$f\left(x\right)=\{x^2,x<1;1+\ln x,x\ge 1\}$

Short answer

The derivative of function is

$$\begin{gathered} f\text{'}\left(x\right)=2x,x<1 \\ DNE,x=1 \\ \frac{1}{x},x>1 \end{gathered}$$

Step-by-step solution

Step 1. Given Information 

The given function is $f\left(x\right)=\{x^2,x<1;1+\ln x,x\ge 1\}$

Step 2. Calculation 

From the differentiation rules,

$$\begin{gathered} \frac{d}{dx}\left(x^2\right)=2x \\ \frac{d}{dx}(1+\ln x)=\frac{1}{x} \\ \mathrm{Let}g\left(x\right)=x^2,h\left(x\right)=1+\ln x \\ \mathrm{And} \\ g\text{'}\left(x\right)=2x,h\text{'}\left(x\right)=\frac{1}{x} \\ \mathrm{Check}g\text{'}\left(1\right)=h\text{'}\left(1\right) \\ g\text{'}\left(1\right)=2\left(1\right) \\ =2 \\ h\text{'}\left(1\right)=\frac{1}{1} \\ =1 \end{gathered}$$

It follows that the function is not continuous at $x=1$.

Thus, the derivative of function is,

$$\begin{gathered} f\text{'}\left(x\right)=2x,x<1 \\ DNE,x=1 \\ \frac{1}{x},x>1 \end{gathered}$$