What is the derivative of x/((x+y^2)) ?

The derivative of x/((x+y^2)) is (y-x-2x\frac{dy)/(dx)}{(x+y)^3}

What is the first derivative of x/((x+y^2)) ?

The first derivative of x/((x+y^2)) is (y-x-2x\frac{dy)/(dx)}{(x+y)^3}

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

\frac{y - x - 2 x \frac{d y}{d x}}{( x + y ) ^{3}}

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

\frac{1 \cdot ( x + y ) ^{2} - 2 ( x + y ) ( 1 + \frac{d y}{d x} ) x}{( ( x + y ) ^{2} ) ^{2}} : \frac{y - x - 2 x \frac{d y}{d x}}{( x + y ) ^{3}}

= \frac{y - x - 2 x \frac{d y}{d x}}{( x + y ) ^{3}}

What is the derivative of x/((x+y^2)) ?
The derivative of x/((x+y^2)) is (y-x-2x\frac{dy)/(dx)}{(x+y)^3}
What is the first derivative of x/((x+y^2)) ?
The first derivative of x/((x+y^2)) is (y-x-2x\frac{dy)/(dx)}{(x+y)^3}