derivative f(x)=5^{sqrt(x)}

解

導関数

f (x) = 5^{x}

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

解答ステップ

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

指数の規則を適用する:

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

5^{x} = e^{x l n (5)}

= \frac{d}{d x} (e^{x l n (5)})

連鎖法則を適用:

e^{x l n (5)} \frac{d}{d x} (x ln (5))

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

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

= e^{x l n (5)} \frac{ln ( 5 )}{2 x}

簡素化

e^{x l n (5)} \frac{ln ( 5 )}{2 x} : \frac{ln ( 5 ) \cdot 5 ^{x}}{2 x}

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