Question

If x² + y² + sin y = 4, then the value of `(d^2y)/(dx^2)` at the point (–2, 0) is ______.

Options

• – 34

• – 32

• – 2

• 4

Answer 1

If x² + y² + sin y = 4, then the value of `(d^2y)/(dx^2)` at the point (–2, 0) is – 34.

Explanation:

Given, x² + y² + sin y = 4

After differentiating the above equation w. r. t. x we get

`2x + 2y (dy)/(dx) + cosy (dy)/(dx)` = 0  ...(i)

`\implies 2x + (2y + cosy) (dy)/(dx)` = 0

`\implies (dy)/(dx) = (-2x)/(2y + cosy)`

At `(-2, 0)(dy/dx)_((-2","  0)) = (-2 xx -2)/(2 xx 0 + cos0)`

`\implies (dy/dx)_((-2","  0)) = 4/(0 + 1)`

`\implies (dy/dx)_((-2","  0))` = 4  ...(ii)

Again differentiating equation (i) w. r. t to x, we get

`2 + 2(dy/dx)^2 + 2y(d^2y)/(dx^2) - siny(dy/dx)^2 + cosy (d^2y)/(dx^2)` = 0

`\implies 2 + (2 - siny)(dy/dx)^2 + (2y + cosy)(d^2y)/(dx^2)` = 0

`\implies (2y + cosy) (d^2y)/(dx^2) = -2 - (2 - sin y)(dy/dx)^2`

`\implies (d^2y)/(dx^2) = (-2 - (2 - siny)(dy/dx)^2)/(2y + cosy)`

So, at (–2, 0),

`(d^2y)/(dx^2) = (-2 - (2 - 0) xx 4^2)/(2 xx 0 + 1)`

`\implies (d^2y)/(dx^2) = (-2 - 2 xx 16)/1`

`\implies (d^2y)/(dx^2)` = – 34