derivative of (x+cos(x)^{cot(x)})

解

\frac{d}{d x} ((x + cos (x))^{c o t (x)})

(- cos (x) + cot (x) - csc^{2} (x) ln (x + cos (x)) (x + cos (x))) (x + cos (x))^{c o t (x) - 1}

解答ステップ

\frac{d}{d x} ((x + cos (x))^{c o t (x)})

指数の規則を適用する:

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

(x + cos (x))^{c o t (x)} = e^{c o t (x) l n (x + c o s (x))}

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

連鎖法則を適用:

e^{c o t (x) l n (x + c o s (x))} \frac{d}{d x} (cot (x) ln (x + cos (x)))

= e^{c o t (x) l n (x + c o s (x))} \frac{d}{d x} (cot (x) ln (x + cos (x)))

\frac{d}{d x} (cot (x) ln (x + cos (x))) = \frac{- cos ( x ) + cot ( x ) - csc ^{2} ( x ) ln ( x + cos ( x ) ) ( x + cos ( x ) )}{x + cos ( x )}

= e^{c o t (x) l n (x + c o s (x))} \frac{- cos ( x ) + cot ( x ) - csc ^{2} ( x ) ln ( x + cos ( x ) ) ( x + cos ( x ) )}{x + cos ( x )}

簡素化

e^{c o t (x) l n (x + c o s (x))} \frac{- cos ( x ) + cot ( x ) - csc ^{2} ( x ) ln ( x + cos ( x ) ) ( x + cos ( x ) )}{x + cos ( x )} : (- cos (x) + cot (x) - csc^{2} (x) ln (x + cos (x)) (x + cos (x))) (x + cos (x))^{c o t (x) - 1}

= (- cos (x) + cot (x) - csc^{2} (x) ln (x + cos (x)) (x + cos (x))) (x + cos (x))^{c o t (x) - 1}