derivative (tan(3x))/(x^2)

解

導関数

\frac{tan ( 3 x )}{x ^{2}}

\frac{3 x sec ^{2} ( 3 x ) - 2 tan ( 3 x )}{x ^{3}}

解答ステップ

\frac{d}{d x} (\frac{tan ( 3 x )}{x ^{2}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( tan ( 3 x ) ) x ^{2} - \frac{d}{d x} ( x ^{2} ) tan ( 3 x )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (tan (3 x)) = sec^{2} (3 x) \cdot 3

\frac{d}{d x} (x^{2}) = 2 x

= \frac{sec ^{2} ( 3 x ) \cdot 3 x ^{2} - 2 x tan ( 3 x )}{( x ^{2} ) ^{2}}

簡素化

\frac{sec ^{2} ( 3 x ) \cdot 3 x ^{2} - 2 x tan ( 3 x )}{( x ^{2} ) ^{2}} : \frac{3 x sec ^{2} ( 3 x ) - 2 tan ( 3 x )}{x ^{3}}

= \frac{3 x sec ^{2} ( 3 x ) - 2 tan ( 3 x )}{x ^{3}}