Pls explain this 2nd derivative?

Question

Pls explain this : x^2 + y^2 = 5 find y'' when x=1 and y=2

Answer 1

x^2 + y^2 = 5

First, implicitly differentiate with respect to x. This requires knowledge of the chain rule, and expressing the derivative of y as y'.

2x + 2y (y') = 0
2y (y') = -2x
y' = [-2x]/[2y]
y' = (-x/y)

Now, differentiate implicitly once again. Note that it's easier to do so this time, with y' isolated.

y'' = [(-1)y - (-x)(y')] / y^2

y'' = [-y + x y'] / (y^2)

Now, substitute y' = (-x/y) in this equation.

y'' = [-y + x(-x/y)] / y^2

Multiply top and bottom by y.

y'' = [-y^2 - x^2] / y^3

When x = 1 and y = 2,

y'' = [-(2)^2 - (1)^2] / [2]^3
y'' = [-4 - 1] / 8
y'' = -5/8

Answer 2

first you have to find the second derivative of y (with respect to x). by for instance implicit differentation. next you fill in the points x = 1 and y = 2, that should give you the value of y''