High School Derivative of Square Root of x at 0

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SUMMARY

The derivative of the function ##x \longmapsto \sqrt{x}## at the point 0 is not defined using either the power rule or the definition of the derivative. The power rule yields 1/2(sqrt. x), which is undefined at 0, while the limit approach leads to infinity. This discrepancy arises because infinity is not a valid output in this context, confirming that the derivative does not exist at x=0.

PREREQUISITES
  • Understanding of calculus concepts, specifically derivatives
  • Familiarity with the power rule for differentiation
  • Knowledge of limits and their properties
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the definition of the derivative in depth
  • Explore the concept of limits, particularly one-sided limits
  • Review the power rule and its applications in calculus
  • Investigate cases of undefined derivatives in calculus
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Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of derivatives and limits in mathematical analysis.

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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