Answer
$f'(x)=3x^2+2x$
Work Step by Step
To take the derivative of a function using the limit process, plug into the equation $f'(x)=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$ and simplify, paying attention to things that cancel out:
$f'(x)=\lim\limits_{h \to 0}\frac{[(x+h)^3+(x+h)^2]-(x^3+x^2)}{h}$
$f'(x)=\lim\limits_{h \to 0}\frac{x^3+2x^2+xh^2+hx^2+2xh^2+h^3+x^2+2xh+h^2-x^3-x^2}{h}$
$f'(x)=\lim\limits_{h \to 0}\frac{2x^2h+xh^2+hx^2+2xh^2+h^3+2xh+h^2}{h}$
$f'(x)=\lim\limits_{h \to 0}2x^2+xh+x^2+2xh+h^2+2x+h$
When you can't simplify any further, plug in $0$ for $h$:
$f'(x)=2x^2+x(0)+x^2+2x(0)+(0)^2+2x+(0)=2x^2+x^2+2x=3x^2+2x$
