Why isn't abs(x) differentiable at x=0? (definition of derivative) | Summary and Q&A

TL;DR

The video explains the concept of differentiability and uses the example of the absolute value function to show that it is continuous but not differentiable at certain points.

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.

Transcript

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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.