derivative of xarccos(1/x)

解

\frac{d}{d x} (x arccos (\frac{1}{x}))

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

解答ステップ

\frac{d}{d x} (x arccos (\frac{1}{x}))

積の法則を適用:

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

f = x, g = arccos (\frac{1}{x})

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

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

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

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

簡素化

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

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