I am currently learning Calculus on my own and I ran into two simple doubts, I am sorry if the doubt is too easy, but bear with me because I am still learning
When do we use the chain rule?
By definition, we are told we use chain rule when differentiating a compound function, being in the format of f(g(x)) Example:
$$f(x) = x²$$ $$g(x) = 2x + 3$$ $$f(g(x)) = (2x + 3)²$$
In the example above it makes sense to use the chain rule, but what if I had another example like this:
$$f(x) = 2x$$ $$g(x) = 3x + 3$$ $$f(g(x)) = 2(3x + 3) = 6x + 6$$
Clearly, in that case, we don't have two use the chain rule, right? Still, it is a compound function, or isn't it?
Is the below notation valid?
$$ f(x) = 2x² + 2 $$ $$ f'(x) = 4x $$ $$F(x) = \int f´(x)dx = f(x)$$ Because sometimes I see:
$$\int f(x)dx $$
And it gets me confused because we are never integrating f(x) but its derivative, right?
Thanks in advance, guys