derivative f(x)=ln|cot(3*x)|

解

導関数

f (x) = ln ∣ cot (3 \cdot x) ∣

- 3 csc^{2} (3 x) tan (3 x)

解答ステップ

\frac{d}{d x} (ln ∣ cot (3 x) ∣)

連鎖法則を適用:

\frac{1}{∣ cot ( 3 x ) ∣} \frac{d}{d x} (∣ cot (3 x) ∣)

= \frac{1}{∣ cot ( 3 x ) ∣} \frac{d}{d x} (∣ cot (3 x) ∣)

\frac{d}{d x} (∣ cot (3 x) ∣) = - \frac{3 csc ^{2} ( 3 x ) cot ( 3 x )}{∣ cot ( 3 x ) ∣}

= \frac{1}{∣ cot ( 3 x ) ∣} (- \frac{3 csc ^{2} ( 3 x ) cot ( 3 x )}{∣ cot ( 3 x ) ∣})

簡素化

\frac{1}{∣ cot ( 3 x ) ∣} (- \frac{3 csc ^{2} ( 3 x ) cot ( 3 x )}{∣ cot ( 3 x ) ∣}) : - 3 csc^{2} (3 x) tan (3 x)

= - 3 csc^{2} (3 x) tan (3 x)