# Chapter 2 - Section 2.8 - The Derivative as a Function - 2.8 Exercises: 43

$f$ is not differentiable at $x=1$ and $x=5$

#### Work Step by Step

There are 3 cases at which a graph is not differentiable at a point: - There is a corner (a pointy shape) at a point in the graph (a pointy point cannot have any tangent lines there) - The graph is not continuous at that point (differentiable means continuous) - There is a vertical tangent line at that point in the graph (since $f'(x)=\infty$) In this graph, there are 2 points at which $f$ is not differentiable there: - At $x=5$, the line of the graph is continuous, but quite vertical. So the tangent line there would be vertical, which means the derivative of $f$ there would be $\infty$. So $f$ is not differentiable there. - At $x=1$, the graph is not continuous. Instead, the graph approaches infinity. Therefore, $f$ is not differentiable there.

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