derivative of (arctanh(sqrt(3x^2))^3)

解

\frac{d}{d x} ((arctanh (3 x^{2}))^{3})

\frac{3 3 x arctanh ^{2} ( 3 x )}{∣ x ∣ ( 1 - 3 x ^{2} )}

解答ステップ

\frac{d}{d x} ((arctanh (3 x^{2}))^{3})

簡素化

(arctanh (3 x^{2}))^{3} : arctanh^{3} (3 x^{2})

= \frac{d}{d x} (arctanh^{3} (3 x^{2}))

連鎖法則を適用:

3 (arctanh (3 x^{2}))^{2} \frac{d}{d x} (arctanh (3 x^{2}))

= 3 (arctanh (3 x^{2}))^{2} \frac{d}{d x} (arctanh (3 x^{2}))

\frac{d}{d x} (arctanh (3 x^{2})) = \frac{3 x}{∣ x ∣ ( 1 - 3 x ^{2} )}

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

簡素化

3 arctanh^{2} (3 x^{2}) \frac{3 x}{∣ x ∣ ( 1 - 3 x ^{2} )} : \frac{3 3 x arctanh ^{2} ( 3 x )}{∣ x ∣ ( 1 - 3 x ^{2} )}

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