derivative of xtan(1/x)

解

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

tan (\frac{1}{x}) - \frac{sec ^{2} ( \frac{1}{x} )}{x}

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

= tan (\frac{1}{x}) - \frac{sec ^{2} ( \frac{1}{x} )}{x}