derivative of arctan^{tan(x}(x))

解

\frac{d}{d x} (arctan^{t a n (x)} (x))

arctan^{t a n (x)} (x) (sec^{2} (x) ln (arctan (x)) + \frac{tan ( x )}{arctan ( x ) ( x ^{2} + 1 )})

解答ステップ

\frac{d}{d x} (arctan^{t a n (x)} (x))

指数の規則を適用する:

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

arctan^{t a n (x)} (x) = e^{t a n (x) l n (a r c t a n (x))}

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

連鎖法則を適用:

e^{t a n (x) l n (a r c t a n (x))} \frac{d}{d x} (tan (x) ln (arctan (x)))

= e^{t a n (x) l n (a r c t a n (x))} \frac{d}{d x} (tan (x) ln (arctan (x)))

\frac{d}{d x} (tan (x) ln (arctan (x))) = sec^{2} (x) ln (arctan (x)) + \frac{tan ( x )}{arctan ( x ) ( x ^{2} + 1 )}

= e^{t a n (x) l n (a r c t a n (x))} (sec^{2} (x) ln (arctan (x)) + \frac{tan ( x )}{arctan ( x ) ( x ^{2} + 1 )})

e^{t a n (x) l n (a r c t a n (x))} = arctan^{t a n (x)} (x)

= arctan^{t a n (x)} (x) (sec^{2} (x) ln (arctan (x)) + \frac{tan ( x )}{arctan ( x ) ( x ^{2} + 1 )})