Question

If y = (tan^–1 x)² then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.

विकल्प

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Answer 1

If y = (tan^–1 x)² then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = 2.

Explanation:

Since, y = (tan^–1 x)² 

`\implies dy/dx = (2tan^-1 x)/(1 + x^2)`

`\implies (1 + x^2) dy/dx = 2tan^-1x = 2sqrt(y)`

`\implies (1 + x^2)^2(dy/dx)^2` = 4y

Again differentiating both sides with respect to x, we get:

`2(1 + x^2)(2x)(dy/dx)^2 + 2(dy/dx)(d^2y)/(dx^2)(1 + x^2)^2 = 4 dy/dx`

`implies 4x(1 + x^2)(dy/dx)^2 + 2(1 + x^2) (dy/dx) (d^2y)/(dx^2) = 4 dy/dx`

`implies 4x(1 + x^2) dy/dx + 2(1 + x^2) (d^2y)/(dx^2)` = 4

`implies (x^2 + 1)^2 (d^2y)/(dx^2) + 2x(1 + x^2) dy/dx` = 2