Understanding the logic behind particular derivative I have 
$\frac{\partial (f(x) g(x))}{\partial x}$=$g(x) f'(x)+f(x) g'(x)$,
I need to differentate this function with respect to x. 
$f(x)=(x+1) (x+2)^2 (x+3)^3 (x+4)^4$
However I do not see the logic using the product rule.

HINT
You must do that step by step.
$f(x)=(x+1) (x+2)^2 (x+3)^3 (x+4)^4$
$$f'(x)=((x+1))' ((x+2)^2 (x+3)^3 (x+4)^4)+((x+1)) ((x+2)^2 (x+3)^3 (x+4)^4)'$$
$$((x+2)^2 (x+3)^3 (x+4)^4)'=((x+2)^2)' ((x+3)^3(x+4)^4)+((x+2)^2) ((x+3)^3(x+4)^4)'$$
$$((x+3)^3(x+4)^4)'=((x+3)^3)'((x+4)^4)+((x+3)^3)((x+4)^4)'$$
now you go from the last to the first.
