What is (\partial)/(\partial x)(x^{-1}y^2+xy^2+5xy) ?

The answer to (\partial)/(\partial x)(x^{-1}y^2+xy^2+5xy) is -(y^2)/(x^2)+y^2+5y

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(\partial)/(\partial x)(x^{-1}y^2+xy^2+5xy)

\frac{\partial}{\partial x} (x^{- 1} y^{2} + x y^{2} + 5 x y)

- \frac{y ^{2}}{x ^{2}} + y^{2} + 5 y

Solution steps

\frac{\partial}{\partial x} (x^{- 1} y^{2} + x y^{2} + 5 x y)

Treat

y

as a constant

Apply the Sum/Difference Rule:

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

= \frac{\partial}{\partial x} (x^{- 1} y^{2}) + \frac{\partial}{\partial x} (x y^{2}) + \frac{\partial}{\partial x} (5 x y)

\frac{\partial}{\partial x} (x^{- 1} y^{2}) = - \frac{y ^{2}}{x ^{2}}

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

\frac{\partial}{\partial x} (5 x y) = 5 y

= - \frac{y ^{2}}{x ^{2}} + y^{2} + 5 y

\int_{1}^{16} \frac{1}{8 - x} d x

\frac{d y}{d x} + 2 x y = y + 4 x - 2

\frac{d}{d x} ((4 x^{3} + 5)^{\frac{3}{2}})

\frac{\partial}{\partial x} (\frac{x ^{2} - 1}{y})

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

What is (\partial)/(\partial x)(x^{-1}y^2+xy^2+5xy) ?
The answer to (\partial)/(\partial x)(x^{-1}y^2+xy^2+5xy) is -(y^2)/(x^2)+y^2+5y