First slide

Derivatives

Question

$If f (x) = \cot^{- 1} (\frac{x^{x} - x^{- x}}{2}) then f^{'} (1) =$

Moderate

A

$- \log 2$
B

$\log 2$
C

1
D

-1

Solution

$\begin{array}{l} f^{'} (x) = \frac{- 1}{1 + {(\frac{x^{x} - x^{- x}}{2})}^{2}} \frac{d}{d x} (\frac{x^{x} - x^{- x}}{2}) \\ = \frac{4}{{(x^{x} + x^{- x})}^{2}} (\frac{x^{x} (1 + \log x) + x^{- x} (1 + \log x)}{2}) \\ f^{'} (1) = \frac{- 4}{(1 + 1)^{2}} \frac{1 (1 + \log 1) + 1 (1 + \log 1)}{2} \\ = - 1 \cdot \frac{1 + 1}{2} \\ = - 1 . \end{array}$

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