derivative of (sqrt(5)^x)

解

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

\frac{1}{2} ln (5) e^{\frac{l n ( 5 ) x}{2}}

解答ステップ

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

簡素化

(5)^{x} : 5^{\frac{x}{2}}

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{x}{2} ln (5)) = \frac{1}{2} ln (5)

= e^{\frac{x}{2} l n (5)} \frac{1}{2} ln (5)

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1}{2} ln (5) e^{\frac{l n ( 5 ) x}{2}}