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Techniques of Differentiation - Derivatives of Natural Logarithmic Functions

Techniques of Differentiation - Derivatives of Natural Logarithmic Functions

Differentiate the following with respect to \(x\) and simplify your answers.

  1. \({{\log }_{3}}\left( {{x}^{2}}\sqrt{{{x}^{3}}+1} \right)\)
  2. \({{\log }_{6}}\left( \frac{\sqrt{x-1}}{3{{x}^{2}}} \right)\)
  3. \({{4}^{1-{{x}^{2}}}}\)
  4. \({{\sin }^{-1}}\left( \ln x \right)\)

Solution:

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Answer:(a) \(\frac{7x^3+4}{2x(x^3+1)\ln3}\), for \(x>-1,\ x\ne0\). (b) \(\frac{4-3x}{2x(x-1)\ln6}\), for \(x>1\). (c) \(-2x\ln4\,4^{1-x^2}\). (d) \(\frac{1}{x\sqrt{1-(\ln x)^2}}\), for \(e^{-1}<x<e\).

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