derivative f(x)=cot^{tan(x)}(x)

解

導関数

f (x) = cot^{t a n (x)} (x)

cot^{t a n (x)} (x) (sec^{2} (x) ln (cot (x)) - sec^{2} (x))

解答ステップ

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

指数の規則を適用する:

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

cot^{t a n (x)} (x) = e^{t a n (x) l n (c o t (x))}

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

連鎖法則を適用:

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

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

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

= e^{t a n (x) l n (c o t (x))} (sec^{2} (x) ln (cot (x)) - sec^{2} (x))

e^{t a n (x) l n (c o t (x))} = cot^{t a n (x)} (x)

= cot^{t a n (x)} (x) (sec^{2} (x) ln (cot (x)) - sec^{2} (x))