derivative of (x+1e^{arctan(x)})

解

\frac{d}{d x} ((x + 1) e^{a r c t a n (x)})

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

解答ステップ

\frac{d}{d x} ((x + 1) e^{a r c t a n (x)})

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x + 1, g = e^{a r c t a n (x)}

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

\frac{d}{d x} (x + 1) = 1

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

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

簡素化

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

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