derivative of 3^{sqrt(x)}

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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