Calc2.60

From Exampleproblems

$f(x)=x^n\,$

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

$=\lim_{\Delta x \to 0}\frac{x^n + n x^{n-1} \Delta x + \frac{n(n-1)}{2!} x^{n-2} \left ( \Delta x \right )^2 + ... + \left ( \Delta x \right )^n - x^n}{\Delta x}$

$=\lim_{\Delta x \to 0} \left [ n x^{n-1} + \frac{n(n-1)}{2!} x^{n-2} \left ( \Delta x \right ) + ... + \left ( \Delta x \right )^{n-1} \right ]$

$= n x^{n-1} \,$