Vtyjkyuk

Please can anyone break down in detailed steps how these 2 lines are equal to each other. I'm confused.

Also please ignore the " = 0 " in the second line.

beginalign*
(1+x^2),&fracdydx+fracd(1+x^2)dxcdot y \
&fracddx,(y(1+x^2))=0
endalign*

Thanks a lot in advance.

The idea here is using the product rule backwards. To see things clearly take $a = 1+ x^2$ and keep $y$ as it is.

We know that from product rule,

$fracd(ay)dx = y fracdadx + a fracdydx$

here the equality is used backwards,

$ y fracdadx + a fracdydx = fracd(ay)dx$

If you are new to product rule and wonder about how it is true, start thinking from the first principles of derivative definition or look at https://en.wikipedia.org/wiki/Product_rule.

$$
(1+x^2)fracdydx+fracd(1+x^2)dxcdot y =fracddx,(y(1+x^2))
$$
Differentiate the right side

You differentiate a product of two functions, just apply the rule
$$fracddx(f times g)= gfracddxf+ffracddxg$$
$$fracddx,(y(1+x^2))=(1+x^2)frac ddxy+yfrac ddx(x^2+1)$$
$$fracddx,(y(1+x^2))=(1+x^2)y'+2xy$$
note that
$$fracddx,(1+x^2)=2x$$