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MHT CET202616 April 2026Evening ShiftMathematicsDifferentiationActual

If [f(x)] = 3x - x^3 1 - 3x^2 and [g(x)] = 1 - x^2 1 + x^2 , then _ x t f(x) - f(t) g(x) - g(t) =

Options

  1. A3 2(1 + t^2)
  2. B3 2
  3. C5 2
  4. D- 5 2(1 + t^2)

Correct answer

B. 3 2

Step-by-step solution

Given [f(x)] = 3x - x^3 1 - 3x^2 Substituting x = , we get: [f(x)] = 3 - ^3 1 - 3 ^2 = (3 ) [f(x)] = ( 2 - 3 ) f(x) = 2 - 3 = 2 - 3 ⁻¹x Differentiating with respect to x , we get: f'(x) = -3 1 + x^2 Similarly, given [g(x)] = 1 - x^2 1 + x^2 Substituting x = , we get: [g(x)] = 1 - ^2 1 + ^2 = (2 ) [g(x)] = ( 2 - 2 ) g(x) = 2 - 2 = 2 - 2 ⁻¹x Differentiating with respect to x , we get: g'(x) = -2 1 + x^2 The given limit is in the form of the definition of the derivative (or can be evaluated using L'Hospital's Rule): _

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