derivative of sqrt(ax^2+|x|^3)

解

\frac{d}{d x} (a x^{2} + ∣ x ∣^{3})

\frac{2 a x ∣ x ∣ + 3 x ^{3}}{2 ∣ x ∣ a x ^{2} + ∣ x ∣ ^{3}}

解答ステップ

\frac{d}{d x} (a x^{2} + ∣ x ∣^{3})

連鎖法則を適用:

\frac{1}{2 a x ^{2} + ∣ x ∣ ^{3}} \frac{d}{d x} (a x^{2} + ∣ x ∣^{3})

= \frac{1}{2 a x ^{2} + ∣ x ∣ ^{3}} \frac{d}{d x} (a x^{2} + ∣ x ∣^{3})

\frac{d}{d x} (a x^{2} + ∣ x ∣^{3}) = 2 a x + \frac{3 x ^{3}}{∣ x ∣}

= \frac{1}{2 a x ^{2} + ∣ x ∣ ^{3}} (2 a x + \frac{3 x ^{3}}{∣ x ∣})

簡素化

\frac{1}{2 a x ^{2} + ∣ x ∣ ^{3}} (2 a x + \frac{3 x ^{3}}{∣ x ∣}) : \frac{2 a x ∣ x ∣ + 3 x ^{3}}{2 ∣ x ∣ a x ^{2} + ∣ x ∣ ^{3}}

= \frac{2 a x ∣ x ∣ + 3 x ^{3}}{2 ∣ x ∣ a x ^{2} + ∣ x ∣ ^{3}}