The book I'm reading introduces derivatives via limits. It gives the following example:
$f(x) = 12x-3x^3$
$f'(x)=\lim{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$
- $=\lim_{\Delta x\rightarrow 0}\frac{12(x+\Delta x)-(x+\Delta x)^3-(12x-x^3)}{\Delta x}$
- $=\lim_{\Delta x\rightarrow 0}\frac{12x+ 12\Delta x -x^3 -3x^2\Delta x -3x(\Delta x)^2-(\Delta x)^3-12x+x^3}{\Delta x}$
- $=\lim_{\Delta x\rightarrow 0}(12 - 3x^2 - 3x\Delta x - (\Delta x)^2)$
- $=12-3x^2$
I'm having trouble with how they got from step 2 to step 3. Where did $\Delta x$ on the bottom go?