What is (\partial)/(\partial y)(xln(x/y)+xy^2) ?

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

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(\partial)/(\partial y)(xln(x/y)+xy^2)

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

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

Solution steps

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

Treat

x

as a constant

Apply the Sum/Difference Rule:

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

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

\frac{\partial}{\partial y} (x ln (\frac{x}{y})) = - \frac{x}{y}

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

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

\int 3 - x d x

15 t^{4} e^{4 t} + 10

6 t \frac{d y}{d t} + 4 y = t

\int e^{x} (sin (e^{x})) d x

\frac{d}{d x} (- 4 cos^{2} (x))

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