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.

Written as in your question $f(x)$ is a product of two functions. In that case you must apply the product rule: $(u(x)v(x))'=u'(x)v(x)+u(x)v'(x)$.
You could also work out the brackets, leading to:$$f(x)=x^4+4x^2+3$$
Written like that $f$ can be recognized as a sum of functions. Then it it is time to use the rule $(u(x)+v(x))'=u'(x)+v'(x)$
So a good thing to ask yourself is: "are we dealing with a product here of with a summation?".
Further there is no rule at all that states that $(u(x)v(x))'=u'(x)v'(x)$. So if that "rule" is part of your luggage then you must throw it away immediately!
