Essential Knowledge 2.2B1

Essential Knowledge 2.2B1 Students will know that a continuous function may fail to be differentiable at a point in its domain.

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As mentioned in Learning Objective 2.2B, trying to take the derivative of a function that isn't differentiable can give misleading answers. So, how do we know if a function is differentiable?

Since most parts of most functions are differentiable, the better question to ask is: when is a function not differentiable.

The core idea to remember is that derivatives are just a special type of limit. If the limit doesn't exist, the derivative doesn't exist. Hold on to this core idea, and you will be able to answer almost anything about differentiability.

However, it's also nice to just have some rules. Here are the 4 cases where a function fails to be differentiable.

1. The curve has a corner

The classic example of this is f(x) = |x|

At x = 0, this curve will have a vertical tangent line. This means the derivative will approach infinity. Remember, strictly speaking, infinite limits do not exist. Therefore the derivative at x = 0 does not exist for this function.

3. The curve is not continuous

Here's a classic example.

You might think, "Wait a tick, the derivative from the left is 0.5, the derivative from the right is 0.5, so the left-handed and right-handed limits agree and the derivative must exist!" That's what I thought, too. So where was I wrong? The limit from the right doesn't actually exist.

There are 2 ways to think about this.

• Just use the difference quotient equation. You'll see that at a = -1, f(a) = -1 but the limit of f(x) as x approaches a = +1, so the limit will be infinite and not exist.

• Imagine the tangent line walking along the curve. When it gets to y = -1, it will have to jump straight up to y = +1, making it, temporarily, a vertical tangent line. We know these mean the derivative is infinite, therefore the limit doesn't exist.