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X^2+y^2+x=0

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Type your math problem... x^2+y^2+x=0

Question

确定圆锥曲线

求圆的标准方程

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求圆的半径

找到圆心

(x + \frac{1}{2})^{2} + y^{2} = \frac{1}{4}

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解方程

求解 x

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求解 y

x = \frac{- 1 + 1 - 4 y ^{2}}{2} x = - \frac{1 + 1 - 4 y ^{2}}{2}

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对称性测试

测试关于原点的对称性

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测试关于 x 轴的对称性

测试关于 y 轴的对称性

Not symmetry with respect to the origin

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求一阶导数

求关于 x 的导数

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求关于 y 的导数

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

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求二阶导数

求出关于 x 的二阶导数

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求出关于 y 的二阶导数

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y ^{2} + 4 x ^{2} + 4 x + 1}{4 y ^{3}}

计算

x^{2} + y^{2} + x = 0

取两边的导数

\frac{d}{d x} (x^{2} + y^{2} + x) = \frac{d}{d x} (0)

计算导数

More Steps

2 x + 2 y \frac{d y}{d x} + 1 = \frac{d}{d x} (0)

计算导数

2 x + 2 y \frac{d y}{d x} + 1 = 0

将表达式移到右侧并更改其符号

2 y \frac{d y}{d x} = 0 - (2 x + 1)

把各项相减

More Steps

2 y \frac{d y}{d x} = - 2 x - 1

两边同时除以

\frac{2 y \frac{d y}{d x}}{2 y} = \frac{- 2 x - 1}{2 y}

把这些数相除

\frac{d y}{d x} = \frac{- 2 x - 1}{2 y}

使用 \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} 重写分数

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

取两边的导数

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

计算导数

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

使用区分规则

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

计算导数

More Steps

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

计算导数

More Steps

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

计算

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y - ( 2 x + 1 ) \times 2 \frac{d y}{d x}}{( 2 y ) ^{2}}

计算

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y - ( 4 x \frac{d y}{d x} + 2 \frac{d y}{d x} )}{( 2 y ) ^{2}}

如果括号外出现负号或减号，请删除括号并更改括号内每个术语的符号

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y - 4 x \frac{d y}{d x} - 2 \frac{d y}{d x}}{( 2 y ) ^{2}}

计算

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y - 4 x \frac{d y}{d x} - 2 \frac{d y}{d x}}{4 y ^{2}}

计算

\frac{d ^{2} y}{d x ^{2}} = - \frac{2 y - 2 x \frac{d y}{d x} - \frac{d y}{d x}}{2 y ^{2}}

使用方程 \frac{d y}{d x} = - \frac{2 x + 1}{2 y} 代入

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y ^{2} + 4 x ^{2} + 4 x + 1}{4 y ^{3}}

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改写方程

r = 0 r = - cos (θ)

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Graph

Graph