What is (\partial)/(\partial x)(x^{-1}y^2+xy^2+3xy) ?

The answer to (\partial)/(\partial x)(x^{-1}y^2+xy^2+3xy) is -(y^2)/(x^2)+y^2+3y

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(\partial)/(\partial x)(x^{-1}y^2+xy^2+3xy)

\frac{\partial}{\partial x} (x^{- 1} y^{2} + x y^{2} + 3 x y)

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

Solution steps

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

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

\int \frac{x ^{2} - x + 5}{x} d x

n = 1 \sum \infty \frac{n}{n + 5}

\frac{d}{d x} (- \frac{4}{3} sin (x) - \frac{5}{3} cos (x))

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

\frac{\partial}{\partial x} (\frac{a x + b y}{( c x + d y )})

What is (\partial)/(\partial x)(x^{-1}y^2+xy^2+3xy) ?
The answer to (\partial)/(\partial x)(x^{-1}y^2+xy^2+3xy) is -(y^2)/(x^2)+y^2+3y