derivative of xarcsin(x)

解答

\frac{d}{d x} (x arcsin (x))

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

求解步骤

\frac{d}{d x} (x arcsin (x))

使用微分乘法定则:

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

f = x, g = arcsin (x)

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

\frac{d x}{d x} = 1

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

= 1 \cdot arcsin (x) + \frac{1}{1 - x ^{2}} x

化简

1 \cdot arcsin (x) + \frac{1}{1 - x ^{2}} x : arcsin (x) + \frac{x}{1 - x ^{2}}

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