derivative of pi^{sqrt(x)}

解

\frac{d}{d x} (π^{x})

\frac{ln ( π ) π ^{x}}{2 x}

解答ステップ

\frac{d}{d x} (π^{x})

指数の規則を適用する:

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

π^{x} = e^{x l n (π)}

= \frac{d}{d x} (e^{x l n (π)})

連鎖法則を適用:

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

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

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

= e^{x l n (π)} \frac{ln ( π )}{2 x}

簡素化

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

= \frac{ln ( π ) π ^{x}}{2 x}