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NCERT Class 12 · Ch 5 Miscellaneous Exercise · Q7

NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Miscellaneous Exercise Question 7

Differentiate \(\log{\left(x \right)}^{\log{\left(x \right)}}\) with respect to x; for x>1.

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: \(\frac{\left(\log{\left(\log{\left(x \right)} \right)} + 1\right) \log{\left(x \right)}^{\log{\left(x \right)}}}{x}\)
  1. Work for x>1.
  2. For a positive base u, write uᵛ=exp(v log u). Its derivative is uᵛ(v′ log u+vu′/u). Here \(\frac{d}{d x} \log{\left(x \right)}^{\log{\left(x \right)}}\) = \(\left(\frac{\log{\left(\log{\left(x \right)} \right)}}{x} + \frac{1}{x}\right) \log{\left(x \right)}^{\log{\left(x \right)}}\).
  3. Combining the terms gives y′ = \(\frac{\left(\log{\left(\log{\left(x \right)} \right)} + 1\right) \log{\left(x \right)}^{\log{\left(x \right)}}}{x}\).

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