Find y'' by implicit differentiation. x2 + 5y2 = 5

Differentiate with respect to x and y

2x + 10 y y' = 0

get y' alone

y' = - 2x / (10 y)

reduce

y'=-x / (5y)

continue with quotient rule of differentiation

y'' = (5y (-1) - (-x) (5y')) / (5y) squared

y'' = (-5y + 5xy') / (25y squared)

reduce by dividing each term by 5

y" = (-y + xy') / (5y squared) answer

Now if the teacher wants you to continue so no y or y' shows

substitute y' = - x / (5y) and y = square root ((5-x squared) / 5)

and simplify