Differentiation Question 4
Question: If $ f(x)={{\cot }^{-1}}( \frac{x^{x}-{x^{-x}}}{2} ), $ then $ f’(1) $ is equal to
[RPET 2000]
Options:
A) - 1
B) 1
C) $ \log 2 $
D) $ -\log 2 $
Show Answer
Answer:
Correct Answer: A
Solution:
$ f(x)={{\cot }^{-1}}( \frac{x^{x}-{x^{-x}}}{2} ) $ ; Put $ x^{x}=\tan \theta $
\ $ y=f(x)={{\cot }^{-1}}( \frac{{{\tan }^{2}}\theta -1}{2\tan \theta } ) $
= $ {{\cot }^{-1}}(-\cot 2\theta ) $ = $ \pi -{{\cot }^{-1}}(\cot 2\theta ) $
Therefore y = $ \pi -2\theta $ = $ \pi -2{{\tan }^{-1}}(x^{x}) $
$ \frac{dy}{dx}=\frac{-2}{1+x^{2x}}.x^{x}(1+\log x) $
Therefore $ {f}’(1)=-1 $ .
BETA
