derivative f(x)=(log_{9}(x))/(7+x^9)

解

導関数

f (x) = \frac{lo g _{9} ( x )}{7 + x ^{9}}

\frac{7 + x ^{9} - 18 ln ( 3 ) x ^{9} lo g _{9} ( x )}{2 ln ( 3 ) x ( 7 + x ^{9} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{lo g _{9} ( x )}{7 + x ^{9}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( lo g _{9} ( x ) ) ( 7 + x ^{9} ) - \frac{d}{d x} ( 7 + x ^{9} ) lo g _{9} ( x )}{( 7 + x ^{9} ) ^{2}}

\frac{d}{d x} (lo g_{9} (x)) = \frac{1}{2 ln ( 3 ) x}

\frac{d}{d x} (7 + x^{9}) = 9 x^{8}

= \frac{\frac{1}{2 l n ( 3 ) x} ( 7 + x ^{9} ) - 9 x ^{8} lo g _{9} ( x )}{( 7 + x ^{9} ) ^{2}}

簡素化

\frac{\frac{1}{2 l n ( 3 ) x} ( 7 + x ^{9} ) - 9 x ^{8} lo g _{9} ( x )}{( 7 + x ^{9} ) ^{2}} : \frac{7 + x ^{9} - 18 ln ( 3 ) x ^{9} lo g _{9} ( x )}{2 ln ( 3 ) x ( 7 + x ^{9} ) ^{2}}

= \frac{7 + x ^{9} - 18 ln ( 3 ) x ^{9} lo g _{9} ( x )}{2 ln ( 3 ) x ( 7 + x ^{9} ) ^{2}}