Chapter 3: Derivatives - Section 3.2 - The Derivative as a Function - Exercises 3.2 - Page 117: 61

See graph and explanations.

Step 1. For $x\gt0$, we have $y=x$ and $y'=\lim_{h\to0}\frac{h+0-0}{h}=1$ Step 2. For $x\lt0$, we have $y=-x$ and $y'=\lim_{h\to0}\frac{-h-0+0}{h}=-1$ Step 3. Graph the function $y'$ as shown. Step 4. Graph the function $y_1=\frac{|x|}{x}$ as shown. Step 5. We conclude that the graphs from step 3 and 4 are identical; thus $y'=y_1=\frac{|x|}{x}$