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The function (f(x)=x^x) is defined for (x>0). What is (f'(e))?

hard MCQ 5 marks Class 11 · Mathematics · Limits and Derivatives
  1. (e^e)
  2. (2e^e)
  3. (e^e(ln e))
  4. (e^e(ln e +1))
Answer
(2e^e)

The correct answer is (2e^e) — Substitute (x=e): (y'=e^e(ln e +1)=e^e(1+1)=2e^e).

Step-by-step solution

  1. Use the logarithmic differentiation: let (y=x^x). Take (ln y=xln x).
  2. Differentiate both sides: (\frac{y'}{y}=1cdotln x + xcdot\frac{1}{x}= ln x + 1).
  3. Solve for (y'): (y'=y(ln x +1)=x^x(ln x +1)).
  4. Substitute (x=e): (y'=e^e(ln e +1)=e^e(1+1)=2e^e).
Common mistake: Forgetting to differentiate (ln x) part correctly

Concept tested: Derivative of (x^x)

Published · How this solution was prepared and checked

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