Verified Solution • 2025–26 Session moderate

QJEEM18Ma429f: JEE Main 2018 Mathematics - Class 12 JEEM18Ma429f Solution: If    f(x) = sin -1 then f' equals :

If    f(x) = sin-1 (2×3x1+9x),\left( {{{2 \times {3^x}} \over {1 + {9^x}}}} \right), then f' (−12)\left( { - {1 \over 2}} \right) equals :

Multiple Choice Options

  • A −3log⁡e3- \sqrt 3 {\log _e}\sqrt 3
  • B 3log⁡e3\sqrt 3 {\log _e}\sqrt 3
  • C −3log⁡e 3- \sqrt 3 {\log _e}\, 3
  • D 3log⁡e 3\sqrt 3 {\log _e}\, 3

Step-by-Step Verified Solution

Since f(x) = sin (2×3x1+9x)\left( {{{2 \times {3^x}} \over {1 + {9^x}}}} \right)

Suppose 3x = tan t

⇒\Rightarrow    f(x) = sin −- 1 (2tan⁡t1+tan⁡2t)\left( {{{2\tan t} \over {1 + {{\tan }^2}t}}} \right) = sin −- 1 (sin2t) = 2t

                                         = 2tan −- 1 (3x)

So, f'(x) = 21+(3x)2×3x.log⁡e3{2 \over {1 + {{\left( {{3^x}} \right)}^2}}} \times {3^x}.{\log _e}3

∴\therefore    f ' (−12)\left( { - {1 \over 2}} \right) = 21+(3−12)2×3−12{2 \over {1 + {{\left( {{3^{ - {1 \over 2}}}} \right)}^2}}} \times {3^{ - {1 \over 2}}} . loge 3

=   12×3×log⁡e3{1 \over 2} \times \sqrt 3 \times {\log _e}3   =   3×log⁡e3\sqrt 3 \times {\log _e}\sqrt 3

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