derivative of 2^{arctan(x})

解

\frac{d}{d x} (2^{a r c t a n (x)})

\frac{ln ( 2 ) \cdot 2 ^{a r c t a n (x)}}{x ^{2} + 1}

解答ステップ

\frac{d}{d x} (2^{a r c t a n (x)})

指数の規則を適用する:

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

2^{a r c t a n (x)} = e^{a r c t a n (x) l n (2)}

= \frac{d}{d x} (e^{a r c t a n (x) l n (2)})

連鎖法則を適用:

e^{a r c t a n (x) l n (2)} \frac{d}{d x} (arctan (x) ln (2))

= e^{a r c t a n (x) l n (2)} \frac{d}{d x} (arctan (x) ln (2))

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

= e^{a r c t a n (x) l n (2)} \frac{ln ( 2 )}{x ^{2} + 1}

簡素化

e^{a r c t a n (x) l n (2)} \frac{ln ( 2 )}{x ^{2} + 1} : \frac{ln ( 2 ) \cdot 2 ^{a r c t a n (x)}}{x ^{2} + 1}

= \frac{ln ( 2 ) \cdot 2 ^{a r c t a n (x)}}{x ^{2} + 1}