What is (\partial)/(\partial x)(x^{-1}y^2+xy^2+4xy) ?

The answer to (\partial)/(\partial x)(x^{-1}y^2+xy^2+4xy) is -(y^2)/(x^2)+y^2+4y

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(\partial)/(\partial x)(x^{-1}y^2+xy^2+4xy)

\frac{\partial}{\partial x} (x^{- 1} y^{2} + x y^{2} + 4 x y)

- \frac{y ^{2}}{x ^{2}} + y^{2} + 4 y

Solution steps

\frac{\partial}{\partial x} (x^{- 1} y^{2} + x y^{2} + 4 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} (4 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} (4 x y) = 4 y

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

\int (1 - x) (6 + x^{2}) d x

sin (x \frac{π}{2})

\int e^{x} \cdot x^{3} d x

y^{^{''}} - 2 y^{^{'}} - 3 y = 3 t e^{2 t}

What is (\partial)/(\partial x)(x^{-1}y^2+xy^2+4xy) ?
The answer to (\partial)/(\partial x)(x^{-1}y^2+xy^2+4xy) is -(y^2)/(x^2)+y^2+4y