derivative of (2^{3x}/(ln(3)))

解

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

\frac{3 ln ( 2 ) \cdot 8 ^{x}}{ln ( 3 )}

解答ステップ

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

定数を除去:

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

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

指数の規則を適用する:

a^{b} = e^{b l n (a)}

2^{3 x} = e^{3 x l n (2)}

= \frac{1}{ln ( 3 )} \frac{d}{d x} (e^{3 x l n (2)})

連鎖法則を適用:

e^{3 x l n (2)} \frac{d}{d x} (3 x ln (2))

= e^{3 x l n (2)} \frac{d}{d x} (3 x ln (2))

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

= \frac{1}{ln ( 3 )} e^{3 x l n (2)} \cdot 3 ln (2)

簡素化

\frac{1}{ln ( 3 )} e^{3 x l n (2)} \cdot 3 ln (2) : \frac{3 ln ( 2 ) \cdot 8 ^{x}}{ln ( 3 )}

= \frac{3 ln ( 2 ) \cdot 8 ^{x}}{ln ( 3 )}