derivative of (3(9^x)/(x^5))

解

\frac{d}{d x} (\frac{3 ( 9 ^{x} )}{x ^{5}})

\frac{3 ^{2 x + 1} ( 2 ln ( 3 ) x - 5 )}{x ^{6}}

解答ステップ

\frac{d}{d x} (\frac{3 ( 9 ^{x} )}{x ^{5}})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 \frac{d}{d x} (\frac{9 ^{x}}{x ^{5}})

商の法則を適用:

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

= 3 \cdot \frac{\frac{d}{d x} ( 9 ^{x} ) x ^{5} - \frac{d}{d x} ( x ^{5} ) 9 ^{x}}{( x ^{5} ) ^{2}}

\frac{d}{d x} (9^{x}) = 2 ln (3) \cdot 9^{x}

\frac{d}{d x} (x^{5}) = 5 x^{4}

= 3 \cdot \frac{2 ln ( 3 ) \cdot 9 ^{x} x ^{5} - 5 x ^{4} \cdot 9 ^{x}}{( x ^{5} ) ^{2}}

簡素化

3 \cdot \frac{2 ln ( 3 ) \cdot 9 ^{x} x ^{5} - 5 x ^{4} \cdot 9 ^{x}}{( x ^{5} ) ^{2}} : \frac{3 ^{2 x + 1} ( 2 ln ( 3 ) x - 5 )}{x ^{6}}

= \frac{3 ^{2 x + 1} ( 2 ln ( 3 ) x - 5 )}{x ^{6}}