What is the derivative of x/(x+y/x) ?

The derivative of x/(x+y/x) is (x(-x\frac{dy)/(dx)+2y)}{(x^2+y)^2}

What is the first derivative of x/(x+y/x) ?

The first derivative of x/(x+y/x) is (x(-x\frac{dy)/(dx)+2y)}{(x^2+y)^2}

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d x}{d x} = 1

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

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

Simplify

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

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

What is the derivative of x/(x+y/x) ?
The derivative of x/(x+y/x) is (x(-x\frac{dy)/(dx)+2y)}{(x^2+y)^2}
What is the first derivative of x/(x+y/x) ?
The first derivative of x/(x+y/x) is (x(-x\frac{dy)/(dx)+2y)}{(x^2+y)^2}