Question

Question #26ad5

Answer 1

`(d^2y)/(dx^2) =-8/y- (64x^2)/y^3`

Explanation:

When we differentiate `y` wrt `x` we get `dy/dx`.

However, we cannot differentiate a non implicit function of `y` wrt `x`. But if we apply the chain rule we can differentiate a function of `y` wrt `y` but we must also multiply the result by `dy/dx`.

When this is done in situ it is known as implicit differentiation.

We have:

`8x^2 + y^2=2`

Differentiate wrt `x`:

`16x + 2ydy/dx = 0`
`:. 8x + ydy/dx = 0`

Differentiate wrt `x` a second time (applying product rule):

`8 + (y)((d^2y)/(dx^2)) + (dy/dx)(dy/dx) = 0`
`:. 8 + y (d^2y)/(dx^2) + (dy/dx)^2 = 0`

and from the earlier equation:

`ydy/dx = -8x => dy/dx = (-8x)/y`

Substituting gives us:

`8 + y (d^2y)/(dx^2) + ((-8x)/y)^2 = 0`
`:. 8 + y (d^2y)/(dx^2) + (64x^2)/y^2 = 0`
`:. y (d^2y)/(dx^2) =-8- (64x^2)/y^2`
`:. (d^2y)/(dx^2) =-8/y- (64x^2)/y^3`