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AP EAMCET202120 Aug 2021Morning ShiftMathematicsDifferentiationActual

If y = 1 + 1 x 1 + 2 x 1 + 3 x ⋯ ⋯ ⋯ 1 + n x and x ≠ 0 . When x = - 1 , d y d x =

Options

  1. An   !
  2. B( n - 1 )   !
  3. C( - 1 ) n ( n - 1 ) !
  4. D( - 1 ) n n !

Correct answer

C. ( - 1 ) n ( n - 1 ) !

Step-by-step solution

Taking log on both sides we get log y = log 1 + 1 x + log 1 + 2 x . . . . . . . . . . . . . log 1 + n x Differentiating, 1 y d y d x = 1 1 + 1 x - 1 x 2 + 1 1 + 2 x - 2 x 2 . . . . . . . . . . . 1 1 + 1 n - n x 2 d y d x = - 1 + 2 x 1 + 3 x . . . . . 1 + n x + 2 1 + 1 x 1 + 2 x . . . . 1 + n x x 2 Now put x = - 1 d y d x = - 1 ( - 1 - 2 - 3 . . . . . . 1 - n ) 1 d y d x = - 1 n 1 2 . . . . . n - 1 = - 1 n n - 1 !

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