Derivative rules I have some problems when I have to derive: I don't know where I have to begin and where I have to stop. For example, if I have $$f(x)=(x^2 +1)(x^2 +3)$$ I know that I have to use the product rule so I get $$f'(x)=(x^2 +1)'(x^2 +3)+(x^2 +1)(x^2 +3)'$$ and the resolution is $$f'(x)=4x^3 +8x$$. But why can't I derive the stuff inside the brackets, like $$f(x)'=(2x)(2x)$$ and then $$f'(x)=4x^2$$
And I always have that problem, I don't know what rule I should use first.
Thank you.

Write $g(x) = x^2+1$ and $h(x) = x^3+3$.
Then your $f$ is
$$ f(x) = g(x)h(x)$$
So the product rule says 
$$
f'(x) = g'(x)h(x) + g(x)h'(x).
$$
The second thing you wrote would be equivalent to 
$$
f'(x) = g'(x)h'(x)
$$
and this is just not how the derivative works.
