derivative (5^{x^3})/(4x^2)

解

導関数

\frac{5 ^{x^{3}}}{4 x ^{2}}

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

解答ステップ

\frac{d}{d x} (\frac{5 ^{x^{3}}}{4 x ^{2}})

定数を除去:

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

= \frac{1}{4} \frac{d}{d x} (\frac{5 ^{x^{3}}}{x ^{2}})

商の法則を適用:

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

= \frac{1}{4} \cdot \frac{\frac{d}{d x} ( 5 ^{x^{3}} ) x ^{2} - \frac{d}{d x} ( x ^{2} ) 5 ^{x^{3}}}{( x ^{2} ) ^{2}}

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

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

= \frac{1}{4} \cdot \frac{3 ln ( 5 ) \cdot 5 ^{x^{3}} x ^{2} x ^{2} - 2 x \cdot 5 ^{x^{3}}}{( x ^{2} ) ^{2}}

簡素化

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

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