## Second derivative

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### Second derivative

What is y'' if xy+y^3=1? The apostrophes stand for second derivative.
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### Re: Second derivative

Do it step-by-step using implicit differentiation
y' means here dy/dx

step 1. diffrentiate both sides

1.1 d(xy +y^3)/dx = d (1)/dx
1.2 y + x d(y)/dx + 3y^2 d(y)/dx = 0 using power rule, composition rule and constant rule
1.3 ( x+ 3y^2) dy/dx = -y

1.4 Hence y' = -y / ( x+ 3y^2)

step 2.From here you can find y", the second derivative of y with respect to x.
Just differentiate both sides. I choose 1.3, but you can also use 1.4 and use Quotient Rule
2.1 (1+6yy' ) y' + (x+3y^2) y'' = -y' Using multiplication rule on the left side.

2.2 Conclusion
y " = -[ 2 y' + +6y(y'^2) ] / ( x + 3y^2) where y' = - y/ ( x+ 3y^2)

No need to replace y' by its value in terms of x and y if no-one asks for it.

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