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

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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

Expand

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

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

\int \frac{x ^{2} - 2 x - 1}{2 x ^{3} + 3 x ^{2} - 2 x} d x

\int \frac{x ^{2}}{e ^{5 x}} d x

\frac{8 e ^{(- 3 s)}}{s ^{3}}

\frac{1}{1 - y}

\frac{d}{d x} (3 x^{2} cos (x) cot (x))

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