derivative of arcsin(3xln(sqrt(x+1)))

解

\frac{d}{d x} (arcsin (3 x) ln (x + 1))

\frac{3 ln ( x + 1 )}{2 1 - 9 x ^{2}} + \frac{arcsin ( 3 x )}{2 ( x + 1 )}

解答ステップ

\frac{d}{d x} (arcsin (3 x) ln (x + 1))

積の法則を適用:

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

f = arcsin (3 x), g = ln (x + 1)

= \frac{d}{d x} (arcsin (3 x)) ln (x + 1) + \frac{d}{d x} (ln (x + 1)) arcsin (3 x)

\frac{d}{d x} (arcsin (3 x)) = \frac{3}{1 - 9 x ^{2}}

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

= \frac{3}{1 - 9 x ^{2}} ln (x + 1) + \frac{1}{2 ( x + 1 )} arcsin (3 x)

簡素化

\frac{3}{1 - 9 x ^{2}} ln (x + 1) + \frac{1}{2 ( x + 1 )} arcsin (3 x) : \frac{3 ln ( x + 1 )}{2 1 - 9 x ^{2}} + \frac{arcsin ( 3 x )}{2 ( x + 1 )}

= \frac{3 ln ( x + 1 )}{2 1 - 9 x ^{2}} + \frac{arcsin ( 3 x )}{2 ( x + 1 )}