Question

How do you find the second derivative of `4x^2 +9y^2 = 36`?

Answer 1

`(d^2y)/(dx^2)=-16/(9y^3)`

Explanation:

differentiate all terms on both sides `color(blue)"implicitly with respect to x"`

`rArr8x+18y.dy/dx=0`

`rArrdy/dx=(-8x)/(18y)=-(4x)/(9y)`

`"to obtain "(d^2y)/(dx^2)" differentiate " dy/dx" using the "color(blue)" quotient rule"`

`(d^2y)/(dx^2)=(9y.(-4)-(-4x).(9dy/dx))/(81y^2)`

`color(white)((d^2y)/(dx^2))=(-36y+36x.dy/dx)/(81y^2)`

`color(white)((d^2y)/(dx^2))=(-36y+36x.(-(4x)/(9y)))/(81y^2)`

`color(white)((d^2y)/(dx^2))=(-36y-(16x^2)/y)/(81y^2)`

`color(white)((d^2y)/(dx^2))=(-36y^2-16x^2)/(81y^3)`

`color(white)((d^2y)/(dx^2))=(-4(9y^2+4x^2))/(81y^3)`

`"now "9y^2+4x^2=36larr" initial statement"`

`rArr(d^2y)/(dx^2)=-144/(81y^3)`

`color(white)(rArr(d^2y)/(dx^2))=-16/(9y^3)`