역함수 미분 극한

$y = \ln ( \tan x )$이면 역함수 $g ( x )$와 관련하여 $x = \ln ( \tan y )$ ∴$\tan ( g ( x ) ) = e ^{x}$, $\displaystyle g ( 0 ) = \frac{\pi}{4}$ $4 g ( 8 h ) - \pi = 4 \left\{ g ( 8 h ) - g ( 0 ) \right\}$이므로 $\displaystyle \lim\limits _{h \rightarrow 0} {\frac{4 \left\{ g ( 8 h ) - g ( 0 ) \right\}}{h}} = \lim\limits _{h \rightarrow 0} {\frac{32 \left\{ g ( 8 h ) - g ( 0 ) \right\}}{8 h} = 32 \times g' ( 0 )}$ $\sec ^{2} ( g ( x ) ) g' ( x ) = e ^{x}$ ∴$\sec ^{2} ( g ( 0 ) ) g' ( 0 ) = 1$ ∴$\displaystyle g' ( 0 ) = \frac{1}{2}$ ∴ 16