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

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

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(\partial)/(\partial x)(y/x+e^{xy}+xy^2)

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

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

Solution steps

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

Treat

y

as a constant

Apply the Sum/Difference Rule:

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

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

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

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

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

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

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