If for x ∈ 0 , 1 4 , the derivative of tan - 1 ⁡ 6 x x 1 - 9 x 3 is x   ⋅ g x , then…

If for x∈0,14, the derivative of tan-1⁡6xx1-9x3 is x ⋅gx , then gx equals:
  1. 91+9x3
  2. 3xx1-9x3
  3. 3x1-9x3
  4. 31+9x3

Solution

Let  3x32=y, then y∈0,38

 tan-1⁡6xx1-9x3=tan-1⁡2y1-y2

⇒ 2tan-1⁡y=2tan-1⁡3x32

Now ⇒ddx2tan-1⁡3x32=21+9x3⋅3⋅32⋅x12

⇒ddx2tan-1⁡3x32=91+9x3 ⋅ x

⇒x.gx=91+9x3 ⋅ x

⇒gx=91+9x3

Asked in: JEE Main 2017 (02 Apr)

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