What is (\partial)/(\partial x)(x^3y-xy^3+sqrt(xy)) ?

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

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(\partial)/(\partial x)(x^3y-xy^3+sqrt(xy))

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

3 y x^{2} - y^{3} + \frac{y}{2 x}

Solution steps

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

Treat

y

as a constant

Apply the Sum/Difference Rule:

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

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

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

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

\frac{\partial}{\partial x} (x y) = \frac{y}{2 x}

= 3 y x^{2} - y^{3} + \frac{y}{2 x}

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

\frac{d}{d x} (9^{2})

y = (1 + 2 x)^{2}, (2, 25)

y^{^{''}} + y^{^{'}} - 2 y = 1 - 2 x^{2}

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