# For what values of x is │x^2-4│ not differentiable?

• gillgill
In summary, differentiability is a property of a function that indicates whether the function has a well-defined derivative at a given point. The absolute value is used in this question because it ensures that the function remains positive and allows for easier analysis of its differentiability. Continuity and differentiability are two separate properties of a function, where continuity refers to the absence of abrupt changes and breaks in the graph. We can determine the values of x for which │x^2-4│ is not differentiable by finding points where the derivative does not exist or where there is a sharp corner or cusp. To prove that a function is not differentiable, we can use the definition of differentiability or look for points of non-differentiability.
gillgill
For what values of x is │x^2-4│ not differentiable?
Is there a way to solve it without looking at the graph?

Yep. Differentiate it and see where the derivative doesn't exist.

You have to use the definition of the derivative (the limit).

Notice how limit from the right and limit from the left are different as x approaches -2 and +2 from both sides.

## 1. What does it mean for a function to be differentiable?

Differentiability is a property of a function that indicates whether the function has a well-defined derivative at a given point. This means that the graph of the function is smooth and has a unique tangent line at that point.

## 2. Why is the absolute value used in this question?

The absolute value is used in this question because the function │x^2-4│ contains a term with an exponent, which could result in a negative value. Using the absolute value ensures that the function remains positive and allows us to analyze its differentiability more easily.

## 3. What is the difference between continuity and differentiability?

Continuity refers to the property of a function where there are no abrupt changes or breaks in the graph. A function can be continuous without being differentiable, but a function cannot be differentiable without being continuous.

## 4. Can we determine the values of x for which │x^2-4│ is not differentiable?

Yes, we can determine these values by finding the points where the derivative of the function does not exist, or where the function has a sharp corner or cusp. These points are also known as points of non-differentiability.

## 5. How can we prove that a function is not differentiable?

To prove that a function is not differentiable, we can use the definition of differentiability and show that the limit of the difference quotient does not exist at a certain point. We can also look for points of non-differentiability, as mentioned in the previous question, and show that the function is not smooth at those points.

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