When does a derivative not exist?
Understand the Problem
The question is asking about the circumstances under which a derivative does not exist in calculus. This can occur under certain conditions like sharp corners, vertical tangents, discontinuities, or points of oscillation in a function.
Answer
A derivative does not exist at points of discontinuity, cusps/corners, vertical tangents, or infinite discontinuities.
A derivative does not exist at a point if the function is not continuous at that point, if there is a cusp or corner at that point, if there is a vertical tangent line at that point, or if the function has a discontinuity.
Answer for screen readers
A derivative does not exist at a point if the function is not continuous at that point, if there is a cusp or corner at that point, if there is a vertical tangent line at that point, or if the function has a discontinuity.
More Information
The concept of where derivatives do not exist is crucial in understanding the behavior of different types of functions.
Tips
A common mistake is to assume a derivative exists because a function appears 'smooth' overall. Always check the specific conditions at the point of interest.
Sources
- Conditions for the existence of a derivative - khanacademy.org
- When derivatives do not exist - mathinsight.org
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