Calc2.55

From Exampleproblems

$f(x)=2x^3\,$

$f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}=\lim_{\Delta x \to 0}\frac{2(x+\Delta x)^3-2x^3}{\Delta x}$

$=\lim_{\Delta x \to 0}\frac{2(x^3+3x^2\Delta x+3x\Delta x^2+\Delta x^3)-2x^3}{\Delta x}$

$=\lim_{\Delta x \to 0}\frac{6x^2\Delta x+6x\Delta x^2+2\Delta x^3}{\Delta x}$

$=\lim_{\Delta x \to 0}6x^2+6x\Delta x+2\Delta x^2=6x^2$

Since the limit exists, we have shown that the derivative of $2 x 3$ is $6 x 2$ .