Find by implicit differentiation.

Question

Find d^2y/dx^2

x^6y^2-4=0

Answer 1

x^6 y^2 - 4 = 0

Later we may need: y^2 = 4/x^6; y = ± 2/(x^3)

Differentiate implicitly:

(x^6 y^2)' - 0 = 0

Use Product Rule:

(x^6)'(y^2) + (x^6)(y^2)' = 0

(6x^5)(y^2) + (x^6)(2y)(y’) = 0

2(x^5)(y)(3y + xy’) = 0

3y + xy’ = 0

Later we may need: y’ = –3y/x.

Differentiate implicitly again:

3y’ + y’ + xy’’ = 0

4y’ + xy’’ = 0

Substitute y’ = –3y/x:

4(–3y/x) + xy’’ = 0

-12y/x + xy’’ = 0

xy’’ = 12y/x

y’’ = 12y/(x^2)

Substitute y = ± 2/(x^3):

y’’ = 12(± 2/(x^3))/(x^2)

y’’ = ± 24/(x^5)