Derivative of Square Root of x at 0

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Discussion Overview

The discussion revolves around the differentiation of the square root function at the point x=0, specifically examining the results obtained through the power rule and the definition of the derivative. The scope includes mathematical reasoning and conceptual clarification regarding limits and undefined values.

Discussion Character

  • Mathematical reasoning, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants note that using the power rule to differentiate the square root function yields 1/2(sqrt. x), which is undefined at x=0.
  • Others argue that applying the definition of the derivative leads to a result of infinity at x=0, raising questions about the discrepancy between the two methods.
  • One participant asserts that the derivative of the function x ↦ √x is not defined at x=0 in either case.
  • A later reply points out that the limit as x approaches 0 from the positive side of 1/(2√x) also approaches infinity.

Areas of Agreement / Disagreement

Participants express disagreement regarding the interpretation of the derivative at x=0, with some asserting it is undefined while others highlight the concept of infinity in the context of the limit.

Contextual Notes

The discussion includes unresolved aspects regarding the definitions and interpretations of limits and derivatives at points of discontinuity or undefined behavior.

mopit_011
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When you use the power rule to differentiate the square root, the result is 1/2(sqrt. x) which is undefined at 0. But, when you use the definition of the definition of the derivative to calculate it, the result is infinity. What causes this difference between these two methods?
 
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Infinity is undefined in this context.
 
mopit_011 said:
When you use the power rule to differentiate the square root, the result is 1/2(sqrt. x) which is undefined at 0. But, when you use the definition of the definition of the derivative to calculate it, the result is infinity. What causes this difference between these two methods?
None. The derivative of ##x\longmapsto \sqrt{x}## isn't defined for ##x=0## in neither case.
 
Some might also observe that
$$\lim_{x\to 0^+} \frac{1}{2\sqrt{x}}$$
Is also infinity.
 
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