IMathAS-libretexts

According to the definition of the derivative,

$\displaystyle \frac{{d}}{{{\left.{d}{x}\right.}}}{\left({x}^{{1}}\right)}$	$\displaystyle =$	$\displaystyle \lim_{{{h}\to{0}}}$ (Enter the simplified form of the difference quotient)
	$\displaystyle =$	(Enter an exponential expression, using negative, fractional, 0, and 1 exponents as appropriate)

$\displaystyle \frac{{d}}{{{\left.{d}{x}\right.}}}{\left({x}^{{2}}\right)}$	$\displaystyle =$	$\displaystyle \lim_{{{h}\to{0}}}$ (Enter the simplified form of the difference quotient)
	$\displaystyle =$	(Enter an exponential expression, using fractional, negative, 0, and 1 exponents as appropriate)

$\displaystyle \frac{{d}}{{{\left.{d}{x}\right.}}}{\left({x}^{{-{1}}}\right)}$	$\displaystyle =$	$\displaystyle \lim_{{{h}\to{0}}}$ (Enter the simplified form of the difference quotient)
	$\displaystyle =$	(Enter an exponential expression, using negative, 0, fractional, and 1 exponents as appropriate)

$\displaystyle \frac{{d}}{{{\left.{d}{x}\right.}}}{\left({x}^{{\frac{{1}}{{2}}}}\right)}$	$\displaystyle =$	$\displaystyle \lim_{{{h}\to{0}}}$ (Enter the simplified form of the difference quotient)
	$\displaystyle =$	(Enter an exponential expression, using negative, 0, 1, and fractional exponents as appropriate)

This suggests

\displaystyle \frac{{d}}{{{\left.{d}{x}\right.}}}{\left({x}^{{n}}\right)}=