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Derivatives
Derivatives
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When Tangent Lines Don't Exist

Tangent lines don't always exist. Since the slope of the tangent line to f at a is the same thing as the derivative of f at a, if f ' (a) doesn't exist then we can't draw a tangent line to f at a.

Sample Problem

Let f(x) = |x|. We saw earlier that f'(0) doesn't exist because of a disagreement between one-sided limits. If we try to draw a tangent line to f at 0, we run into difficulties. Should it look like this?

Or should it look like this?

Since there's no way to decide which it should be, we can't draw the tangent line to f at 0 at all.

Sample Problem

We can't draw a tangent line to f at a for the function f shown below:

Again, it's because of a disagreement between the one-sided limits. If we approach a from the right, we think the tangent line should look like this:

But if we approach a from the left, we think the tangent line should look like this:

In general, we can't draw tangent lines at parts of a graph that look like "points" or "corners":

We also can't draw tangent lines at places where the function doesn't exist.

This makes sense if we think about the limit definition of the derivative. We need to use f(a) to calculate f ' (a), so if f(a) doesn't exist, we're out of luck.

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