derivative of sqrt(x)tan(sqrt(3/x))

解

\frac{d}{d x} (x tan (\frac{3}{x}))

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

解答ステップ

\frac{d}{d x} (x tan (\frac{3}{x}))

積の法則を適用:

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

f = x, g = tan (\frac{3}{x})

= \frac{d}{d x} (x) tan (\frac{3}{x}) + \frac{d}{d x} (tan (\frac{3}{x})) x

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

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

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

簡素化

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

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