This picture shows an animation of the derivative at different points along a continuous line with a sharp corner. The derivative exists everywhere except at the point x=4 corresponding to the corner.

The derivative does not exist at the value of x=4 where the corner occurs since the slope of the tangent line would be different depending on if you approached the corner from the left or the right. At this point, the tangent line is not smoothly varying.

This is discussed in the text on page 172.