derivative of sqrt(x)sec(1/(sqrt(x)))

解

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

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

解答ステップ

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

積の法則を適用:

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

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

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

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

\frac{d}{d x} (sec (\frac{1}{x})) = - \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} )}{2 x ^{\frac{3}{2}}}

= \frac{1}{2 x} sec (\frac{1}{x}) + - \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} )}{2 x ^{\frac{3}{2}}} x

簡素化

\frac{1}{2 x} sec (\frac{1}{x}) + - \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} )}{2 x ^{\frac{3}{2}}} x : \frac{sec ( \frac{1}{x} )}{2 x} - \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} )}{2 x}

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