2x^2+y^2=5 Find y'' (Double Prime)

2x² + y² = 5

Find y' using implicit differentiation:

4x + 2y y' = 0
2y y' = −4x
y' = −4x/(2y)
y' = −2x/y

Find y'' using quotient rule:

y'' = (−2 * y − (−2x) y') / y²
y'' = (−2y + 2x y') / y²
y'' = (−2y + 2x (−2x/y)) / y²
y'' = (−2y − 4x²/y) / y²
y'' = −2/y (y² + 2x²) / y²
y'' = −2 (2x² + y²) / y³

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Alternate method:
y' = −2x/y
y y' = −2x

Differentiate both sides:

y' * y' + y * y'' = −2
(−2x/y)² + y y'' = −2
4x²/y² + y y'' = −2
4x² + y³ y'' = −2y²
y³ y'' = −4x² − 2y²
y³ y'' = −2 (2x² + y²)
y'' = −2 (2x² + y²) / y³

y^2 = -2x^2+5
y' = -4x
y" = -4