derivative f(x)=(xtan^2(x))/(1+x^2)

解

導関数

f (x) = \frac{x tan ^{2} ( x )}{1 + x ^{2}}

\frac{2 x ^{3} sec ^{2} ( x ) tan ( x ) - x ^{2} tan ^{2} ( x ) + 2 x sec ^{2} ( x ) tan ( x ) + tan ^{2} ( x )}{( 1 + x ^{2} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x tan ^{2} ( x )}{1 + 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} ( x tan ^{2} ( x ) ) ( 1 + x ^{2} ) - \frac{d}{d x} ( 1 + x ^{2} ) x tan ^{2} ( x )}{( 1 + x ^{2} ) ^{2}}

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

\frac{d}{d x} (1 + x^{2}) = 2 x

= \frac{( tan ^{2} ( x ) + 2 x sec ^{2} ( x ) tan ( x ) ) ( 1 + x ^{2} ) - 2 xx tan ^{2} ( x )}{( 1 + x ^{2} ) ^{2}}

簡素化

\frac{( tan ^{2} ( x ) + 2 x sec ^{2} ( x ) tan ( x ) ) ( 1 + x ^{2} ) - 2 xx tan ^{2} ( x )}{( 1 + x ^{2} ) ^{2}} : \frac{2 x ^{3} sec ^{2} ( x ) tan ( x ) - x ^{2} tan ^{2} ( x ) + 2 x sec ^{2} ( x ) tan ( x ) + tan ^{2} ( x )}{( 1 + x ^{2} ) ^{2}}

= \frac{2 x ^{3} sec ^{2} ( x ) tan ( x ) - x ^{2} tan ^{2} ( x ) + 2 x sec ^{2} ( x ) tan ( x ) + tan ^{2} ( x )}{( 1 + x ^{2} ) ^{2}}