What is (\partial)/(\partial x)((x^2y+3zy^3)/(xyz)) ?

The answer to (\partial)/(\partial x)((x^2y+3zy^3)/(xyz)) is (x^2-3y^2z)/(x^2z)

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(\partial)/(\partial x)((x^2y+3zy^3)/(xyz))

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

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

Solution steps

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

Treat

y, z

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{yz} \frac{\partial}{\partial x} (\frac{x ^{2} y + 3 z y ^{3}}{x})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{1}{yz} \cdot \frac{\frac{\partial}{\partial x} ( x ^{2} y + 3 z y ^{3} ) x - \frac{\partial}{\partial x} ( x ) ( x ^{2} y + 3 z y ^{3} )}{x ^{2}}

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

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

= \frac{1}{yz} \cdot \frac{2 y xx - 1 \cdot ( x ^{2} y + 3 z y ^{3} )}{x ^{2}}

Simplify

\frac{1}{yz} \cdot \frac{2 y xx - 1 \cdot ( x ^{2} y + 3 z y ^{3} )}{x ^{2}} : \frac{x ^{2} - 3 y ^{2} z}{x ^{2} z}

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

x y + y^{^{'}} = 80 x

\int_{3}^{4} (4 + \frac{24}{x ^{2}}) d x

\frac{d y}{d x} + 7 x y = 0

x \to 2 lim (x^{3} - x^{2} - 4)

\frac{d}{d t} (3 t - t^{3})

What is (\partial)/(\partial x)((x^2y+3zy^3)/(xyz)) ?
The answer to (\partial)/(\partial x)((x^2y+3zy^3)/(xyz)) is (x^2-3y^2z)/(x^2z)