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Question: If f(x) = cos<sup>–1</sup>\(\left\lbrack \frac{1 - \left( \log x \right)^{2}}{1 + \left( \log x \rig...

If f(x) = cos–1[1−(log⁡x)21+(log⁡x)2]\left\lbrack \frac{1 - \left( \log x \right)^{2}}{1 + \left( \log x \right)^{2}} \right\rbrack, then f ¢(5) =

A

Does not exist

B

2/e

C

1/e

D

1

Answer

1/e

Explanation

Solution

f(x) = 2 tan–1(log x)

f(x) = 21+(log⁡x)2\frac{2}{1 + (\log x)^{2}} × 1x\frac{1}{x}

f´(5) = 21+(log⁡e)2\frac{2}{1 + (\log e)^{2}} × 1e\frac{1}{e} = 22\frac{2}{2}×1e\frac{1}{e} = 1e\frac{1}{e}