JEE Maths - Differentiation

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Question - 1

If the derivative of f(tan x) w.r.t. g(sec x) at x = \(\pi\)/4,where f '(1) = 2 and. g'(\(\sqrt{2}\)) = 4 is \(\frac{\sqrt{2}}{k}\) then find k ?

  • A 1.00
  • B 2.00
  • C 3.00
  • D 4.00

Question - 2

Let \(\phi\)(x) be the inverse of the function f(x) and f '(x) = \(\frac{1}{1+x^{5}} \text { and } \frac{d}{d x} \phi(x)=[1+\phi(x)]^{n}\) then find the n. 

  • A 1.00
  • B 3.00
  • C 5.00
  • D 7.00

Question - 3

Let y be a function of x, such that log (x + y) - 2xy = 0, then find y' (0).

  • A 1.00
  • B 2.00
  • C 3.00
  • D 4.00

Question - 4

Let f(x), g(x) be two continuously differentiable functions satisfying the relationships f '(x) = g(x)
and f'(x) = -f(x).Let h(x) = [f(x)2 + [g(x)]2. If h(0) = 5, then find value of h(10).

  • A 1.00
  • B 5.00
  • C 6.00
  • D 8.00

Question - 5

If 5 f (x) + 3 f (1/x) = x + 2,then find \(\frac{d}{dx}\)(x.f(x)) = 1.

  • A 0.87
  • B 0.67
  • C 0.37
  • D 0.57

Question - 6

If y = tan-1 \(\frac{2^x}{1+2^{2x+1}}\), then \(\frac{dy}{dx}\) at x = 0 is k log \(\frac{1}{2}\) then find k. 

  • A 0.60
  • B 0.30
  • C 0.10
  • D 0.20

Question - 7

Find the first derivative of the function \(\left[\cos ^{-1}\left(\sin \sqrt{\frac{1+x}{2}}\right)+x^{x}\right]\) with respect to x at x = 1.

  • A 0.25
  • B 0.45
  • C 0.75
  • D 0.95

Question - 8

Let g(x) be inverse of function f (x) = x3 + 2x2 + 4x + sin \([\frac{\pi}{2}x]\) and g'(8) is equal to \(\frac{1}{k}\) then find k.

  • A 11.00
  • B 21.00
  • C 31.00
  • D 41.00

Question - 9

Find \(\frac{d}{d x}\left[\sin ^{2} \cot ^{-1}\left\{\sqrt{\frac{1-x}{1+x}}\right\}\right]\).

  • A 0.25
  • B 0.50
  • C 0.75
  • D 0.100

Question - 10

If y = \(\tan ^{-1} \frac{\log \left(e / x^{2}\right)}{\log \left(e x^{2}\right)}+\tan ^{-1} \frac{3+2 \log x}{1-6 \log x}\) , then find \(\frac{d^2y}{dx^2}\).

  • A 0.00
  • B 1.00
  • C 2.00
  • D 3.00
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