Are you looking for d²y /dx² ? Or further derivatives?

the second derivative ! thank you

dy/dx = -3y / √(1 - 9x²)


√(1 - 9x²) dy/dx = -3y

(1 - 9x²)¹/² dy/dx = -3y


Differentiate both sides with respect to x,

½(1 - 9x²)-¹/² (-18x) dy/dx + (1 - 9x²)¹/² d²y/dx² = -3 dy/dx

(-9x dy/dx) / √(1 - 9x²) + √(1 - 9x²) d²y/dx² = -3 dy/dx


√(1 - 9x²) d²y/dx² = (9x dy/dx) / √(1 - 9x²) - 3 dy/dx


√(1 - 9x²) d²y/dx² = 9x(-3y / √(1 - 9x²)) / √(1 - 9x²) - 3(-3y / √(1 - 9x²)


√(1 - 9x²) d²y/dx² = -27xy / (1 - 9x²) + 9y / √(1 - 9x²)


d²y/dx² = 9y / (1 - 9x²) - 27xy / (1 - 9x²)³/²

d²y/dx² = 9y√(1 - 9x²) / (1 - 9x²)³/² - 27xy / (1 - 9x²)³/²

d²y/dx² = 9y(√(1 - 9x²) - 3x) / (1 - 9x²)³/²

Alternatively , use quotient rule.


dy/dx = -3y / √(1 - 9x²) = -3y / (1 - 9x²)¹/²


Differentiate both sides with respect to x,


d²y/dx² = [(1 - 9x²)¹/² (-3dy/dx) - ½(1 - 9x²)-¹/² (-18x) (-3y)] / ((1 - 9x²)¹/²)²


= [-3dy/dx (1 - 9x²)¹/² - 27xy(1 - 9x²)-¹/²] / (1 - 9x²)


= [-3(-3y / (1 - 9x²)¹/²) (1 - 9x²)¹/² - 27xy(1 - 9x²)-¹/²] / (1 - 9x²)

= [9y - 27xy(1 - 9x²)-¹/²] / (1 - 9x²)

= [9y(1 - 9x²)¹/² - 27xy] / (1 - 9x²)³/²

= 9y[√(1 - 9x²) - 3x] / √(1 - 9x²)³


This way is actually much easier.