Q102 / 160 · AP EAMCET 2024 (20 May Shift 1) · Mathematics

If $y=\tan (\log x)$, then $\frac{d^2 y}{d x^2}=$

  • A
    $\frac{-\sec ^2(\log x)[1+2 \tan x]}{x^2}$
  • B
    $\frac{\sec ^2(\log x)[1+\tan (\log x)]}{x^2}$
  • C
    $\frac{\sec (\log x)[2+\tan (\log x)-1]}{x^2}$
  • Correct answer
    D
    $\frac{\sec ^2(\log x)[2+\tan (\log x)-1]}{x^2}$

Solution

$y=\tan (\log x)$ $\frac{d y}{d x}=\frac{\sec ^2(\log x)}{x}$ $\frac{d^2 y}{d x^2}=\frac{2 \sec ^2(\log x) \tan (\log x)}{x^2}-\frac{\sec ^2(\log x)}{x^2}$ $=\frac{\sec ^2(\log x)}{x^2}[2+\tan (\log x)-1]$

Hence, the correct answer is option: 'D'.

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