derivative f(x)=(tan(1/x))^{sec(1/x)}

解答

导数

f (x) = (tan (\frac{1}{x}))^{s e c (\frac{1}{x})}

tan^{s e c (\frac{1}{x})} (\frac{1}{x}) (- \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} ) ln ( tan ( \frac{1}{x} ) )}{x ^{2}} - \frac{sec ^{3} ( \frac{1}{x} )}{x ^{2} tan ( \frac{1}{x} )})

求解步骤

\frac{d}{d x} ((tan (\frac{1}{x}))^{s e c (\frac{1}{x})})

使用指数法则:

a^{b} = e^{b l n (a)}

(tan (\frac{1}{x}))^{s e c (\frac{1}{x})} = e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))}

= \frac{d}{d x} (e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))})

使用连锁律:

e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))} \frac{d}{d x} (sec (\frac{1}{x}) ln (tan (\frac{1}{x})))

= e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))} \frac{d}{d x} (sec (\frac{1}{x}) ln (tan (\frac{1}{x})))

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

= e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))} (- \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} ) ln ( tan ( \frac{1}{x} ) )}{x ^{2}} - \frac{sec ^{3} ( \frac{1}{x} )}{x ^{2} tan ( \frac{1}{x} )})

e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))} = tan^{s e c (\frac{1}{x})} (\frac{1}{x})

= tan^{s e c (\frac{1}{x})} (\frac{1}{x}) (- \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} ) ln ( tan ( \frac{1}{x} ) )}{x ^{2}} - \frac{sec ^{3} ( \frac{1}{x} )}{x ^{2} tan ( \frac{1}{x} )})

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