Question

How do you find `(d^2y)/(dx^2)` for `2=2x^2-4y^2`?

Answer 1

`(d^2y)/(dx^2)=(2y^2-x^2)/(4y^3)`

Explanation:

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

`rArr0=4x-8y.dy/dx`

`rArrdy/dx=(-4x)/(-8y)=x/(2y)`

The second derivative is obtained by differentiating `dy/dx`

differentiate `dy/dx" using the " color(blue)"quotient rule"`

`rArr(d^2y)/(dx^2)=(2y.1-x.2dy/dx)/(4y^2)`

`color(white)(rArr(d^y)/(dx^2))=(2y-2x(x/(2y)))/(4y^2)`

`color(white)(rArr(d^2y)/(dx^2))=(2y-((x^2)/y))/(4y^2)`

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