What is the derivative of sqrt(ax^2+|x|^3) ?

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

What is the first derivative of sqrt(ax^2+|x|^3) ?

The first derivative of sqrt(ax^2+|x|^3) is (2ax|x|+3x^3)/(2|x|sqrt(ax^2+|x|^3))

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\frac{d}{d x} (a x^{2} + ∣ x ∣^{3})

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

Solution steps

\frac{d}{d x} (a x^{2} + ∣ x ∣^{3})

Apply the chain rule:

\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 ∣})

Simplify

\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}}

\frac{\partial}{\partial z} (x cos (\frac{y}{z}))

\frac{d}{d x} (c_{1} e^{- 4 x} + c_{2} x e^{- 4 x})

\frac{\partial}{\partial y} (x^{4} + y^{4})

\int \frac{1}{3} sin (x) d x

\int 21 tan^{3} (θ) d θ

What is the derivative of sqrt(ax^2+|x|^3) ?
The derivative of sqrt(ax^2+|x|^3) is (2ax|x|+3x^3)/(2|x|sqrt(ax^2+|x|^3))
What is the first derivative of sqrt(ax^2+|x|^3) ?
The first derivative of sqrt(ax^2+|x|^3) is (2ax|x|+3x^3)/(2|x|sqrt(ax^2+|x|^3))