What is (\partial)/(\partial x)(x+y+sqrt(|x|^3y^2)) ?

The answer to (\partial)/(\partial x)(x+y+sqrt(|x|^3y^2)) is (3xy)/(2sqrt(|x|))+1

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(\partial)/(\partial x)(x+y+sqrt(|x|^3y^2))

\frac{\partial}{\partial x} (x + y + ∣ x ∣^{3} y^{2})

\frac{3 x y}{2 ∣ x ∣} + 1

Solution steps

\frac{\partial}{\partial x} (x + y + ∣ x ∣^{3} y^{2})

Simplify

x + y + ∣ x ∣^{3} y^{2} : x + y + y^{2} ∣ x ∣^{3}

= \frac{\partial}{\partial x} (x + y + y^{2} ∣ x ∣^{3})

Treat

y

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial x} (x) + \frac{\partial}{\partial x} (y) + \frac{\partial}{\partial x} (y^{2} ∣ x ∣^{3})

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (y) = 0

\frac{\partial}{\partial x} (y^{2} ∣ x ∣^{3}) = \frac{3 y x}{2 ∣ x ∣}

= 1 + 0 + \frac{3 y x}{2 ∣ x ∣}

Simplify

= \frac{3 x y}{2 ∣ x ∣} + 1

\frac{d}{d x} (5 x^{2 π})

d er i v a t i v e (\frac{10 - ( 2 )}{( x )}) \cdot (\frac{3 + ( 1 )}{( x )})

(\frac{1}{1 + e ^{x}})^{^{'}}

d er i v a t i v e - 3 x x + 1

x \to π lim (\frac{e ^{3 x} - 1}{x})

What is (\partial)/(\partial x)(x+y+sqrt(|x|^3y^2)) ?
The answer to (\partial)/(\partial x)(x+y+sqrt(|x|^3y^2)) is (3xy)/(2sqrt(|x|))+1