What is (\partial)/(\partial x)(5/(xy)+x/y+y/x) ?

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

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(\partial)/(\partial x)(5/(xy)+x/y+y/x)

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

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

Solution steps

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

Treat

y

as a constant

Apply the Sum/Difference Rule:

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

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

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

\frac{\partial}{\partial x} (\frac{x}{y}) = \frac{1}{y}

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

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

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