derivative (0.09(5^x))/(x^3)

解

導関数

\frac{0.09 ( 5 ^{x} )}{x ^{3}}

\frac{0.09 \cdot 5 ^{x} ( 1.60943 \dots x - 3 )}{x ^{4}}

解答ステップ

\frac{d}{d x} (\frac{0.09 \cdot 5 ^{x}}{x ^{3}})

定数を除去:

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

= 0.09 \frac{d}{d x} (\frac{5 ^{x}}{x ^{3}})

商の法則を適用:

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

= 0.09 \cdot \frac{\frac{d}{d x} ( 5 ^{x} ) x ^{3} - \frac{d}{d x} ( x ^{3} ) 5 ^{x}}{( x ^{3} ) ^{2}}

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

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

= 0.09 \cdot \frac{5 ^{x} ln ( 5 ) x ^{3} - 3 x ^{2} \cdot 5 ^{x}}{( x ^{3} ) ^{2}}

簡素化

0.09 \cdot \frac{5 ^{x} ln ( 5 ) x ^{3} - 3 x ^{2} \cdot 5 ^{x}}{( x ^{3} ) ^{2}} : \frac{0.09 \cdot 5 ^{x} ( 1.60943 \dots x - 3 )}{x ^{4}}

= \frac{0.09 \cdot 5 ^{x} ( 1.60943 \dots x - 3 )}{x ^{4}}