Consider the following statements: I. \(\dfrac{d}{dx}\ln|x| = -\dfrac{1}{x}\) if \(x < 0\) II. \(\dfrac{d}{dx}\ln|x| = \dfrac{1}{x}\) if \(x > 0\) Which of the statements given above is/are correct?
II only
The correct derivative of \(\ln|x|\) is \(\dfrac{1}{x}\) for all \(x \neq 0\), which can be verified by writing \(\ln|x| = \ln(-x)\) for \(x < 0\) and differentiating using the chain rule to get \(\dfrac{-1}{-x} = \dfrac{1}{x}\). So statement I, which claims the derivative is \(-\dfrac{1}{x}\) for \(x < 0\), is incorrect, while statement II, which gives \(\dfrac{1}{x}\) for \(x > 0\), is correct. Hence only statement II is correct.
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Select the correct answer using the code given below
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\({\rm{f}}\left( {\rm{x}} \right) = \left| {\begin{array}{*{20}{c}} {{{\rm{x}}^3}}&{\sin {\rm{x}}}&{\cos {\rm{x}}}\\ 6&{ - 1}&0\\ {\rm{p}}&{{{\rm{p}}^2}}&{{{\rm{p}}^3}} \end{array}} \right|\) , where p is a constant
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