Trouble understanding Taylor polynomial

Hi There,

Below is an answer from one of my assignments. I am trying to understand the answer however i am stuck on the parts as listed below.

I have been through my books and online a thousand time but i just cant get my head around this. Your help will be very much appreciated!

Rearrange to the form

y′ = 2x2 − y2 (1)

and evaluate at x = 0:

y′(0) = −y2(0) = −4 .

Differentiate (1): How does this differentiate?

y′′ = 4x − 2yy′ (2)

from where

y′′(0) = −2y(0)y′(0) = 16 .

Now differentiate (2):

y′′′ = 4 − 2(y′)2 − 2yy′′ (3) And this one also?

from where

y′′′(0) = 4 − 2(y′(0))2 − 2y(0)y′′(0) = −92 .

Differentiate (3):

y′′′′ = −4y′y′′ − 2y′y′′ − 2yy′′′ (4) And again?