In my studies, I encountered a question which I did not know how to do. The answer referred to implicit differentiation but I did not understand the steps undertaken, so could someone explain to me how to do the following question?

The answer states that $\displaystyle \frac{d^2y}{dx^2} = \frac{d}{dx}(\frac{dy}{dx}) = \frac{d(y-y^2)}{dy} \cdot y(1-y) = (1-2y)y(1-y)$. Where does the $\displaystyle y - y^2$ come from?$\displaystyle \frac{dy}{dx}=y(1-y)$, show that $\displaystyle \frac{d^2y}{dx^2} = (1-2y)y(1-y)$.

Then I had problem with Euler's method with the question:

How do I use Euler's method if all I have if $\displaystyle f(y)$?Given that $\displaystyle y=2$ when $\displaystyle x=0$, use Euler's method with a step-size of $\displaystyle \frac{1}{4}$ to estimate the value of $\displaystyle y$ when $\displaystyle x = \frac{1}{2}$