Basic math problem with Integrating Factors: Differential Equations I was watching a youtube tutorial on Integrating Factors and I'm lost in a part where the derivative of: xy' + 1y becomes xy.
Please I need some clarification on that part.
Secondly.
I was watching a another youtube tutorial on Integrating Factors and I also got lost in this part.
https://i.stack.imgur.com/pmcP6.png
I don't understand how the first line converts to the other, Please I also need some clarification here.
Thanks.

Suppose you have 
$$xy'+1y=0$$
Notice that we have 
this is that we have $$\frac{d}{dx}(xy)=x\frac{dy}{dx}+\frac{dx}{dx}y=xy'+y$$
by product rule.
Similarly, for $$\frac{dy}{dx}+\frac{2x}{1+x^2}y=0$$
By multiplying integrating factor of $(1+x^2)$, we have
$$(1+x^2) \frac{dy}{dx}+2xy=0$$
which can be written as 
$$(1+x^2) \frac{dy}{dx}+\frac{d(1+x^2)}{dx}\cdot y$$
or 
$$\frac{d}{dx}(y(1+x^2))=0$$
