Why Is the Absolute Value Function Not Differentiable at Zero?

TL;DR

The absolute value function is not differentiable at x=0 because the limits from the left and right do not match; the right limit equals 1 while the left limit equals -1. This discrepancy indicates that a slope (derivative) cannot be defined at this point, despite the function being continuous.

Transcript

so first we'll take a look at the picture this right here is the graph for absolute x and when x is equal to zero it's this point at this corner and remember differentiable means that we should be able to find the derivative and derivative means what the slope of the tangent line right and now the question is can we draw tangent line at this corner... Read More

Key Insights

• 😥 Differentiability means being able to find the derivative of a function at every point.
• 📈 The absolute value function is not differentiable at corners or sharp turns on its graph.
• 👍 Proving the differentiability of a function requires using the definition of the derivative and evaluating limits.

Questions & Answers

Q: What does it mean for a function to be differentiable?

Differentiability means that the derivative of the function, which represents the slope of the tangent line, can be determined at every point of the function.

Q: Why is the absolute value function not differentiable at corners or sharp turns?

At corners or sharp turns on the graph of the absolute value function, it is not possible to draw a tangent line that represents the slope of the function because the slope changes abruptly.

Q: How can the differentiability of a function be proven mathematically?

The differentiability of a function can be proven by using the definition of the derivative. By taking the limit of the function as it approaches a specific point, it can be determined whether the derivative exists at that point or not.

Q: What is the significance of the derivative in relation to differentiability?

The derivative represents the slope of the tangent line at a specific point on a function. If the derivative does not exist at a certain point, the function is not differentiable at that point.

Summary & Key Takeaways

• The video discusses the graph of the absolute value function and explains that differentiability means being able to find the derivative, which represents the slope of the tangent line.

• It is demonstrated that the absolute value function is not differentiable at corners or sharp turns on the graph.

• The video also shows how to mathematically prove that the absolute value function is not differentiable using the definition of the derivative.