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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Simplify

\frac{1 \cdot ( x + \frac{y}{x} ) - ( 1 - \frac{y}{x ^{2}} ) x}{( x + \frac{y}{x} ) ^{2}} : \frac{2 x y}{( x ^{2} + y ) ^{2}}

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

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