calculus problem using second derivative?

Question

If x^2 - 2y^2 = 2, then (d^2 y)/(d x^2) can be expressed in the form a y^b, where a and b are rational numbers in simplest form.

find a and b.

an explanation would be great

Answer 1

We start by using implicit differentiation:

x² - 2y² = 2
2x - 4y dy/dx = 0
dy/dx = x/(2y)

Differentiating a second time:

d²y/dx² = (2y - 2 dy/dx x)/(4y²)

Substituting the value we already have for dy/dx:

d²y/dx² = (2y - 2 x/(2y) x)/(4y²)
d²y/dx² = (2y - x²/y)/(4y²)

Solving the original equation for x²:

x² - 2y² = 2
x²=2+2y²

And substituting:

d²y/dx² = (2y - (2+2y²)/y)/(4y²)
d²y/dx² = (2y - 2/y - 2y)/(4y²)
d²y/dx² = (-2/y)/(4y²)
d²y/dx² = -1/(2y³)

d²y/dx² = -1/2 * y^(-3)

so a=-1/2 and b=-3

Answer 2

a = -1/2
b = -3

Let me know if you need an explanation...