![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/pmathc202a.svg\")
<\/p>\n<\/p>\n
\\(f(x, y) = x^2 + y^2 -1 = 0\\) \u3092\u6e80\u305f\u3059\u9670\u95a2\u6570 \\(y=y(x)\\) \u306e\u6700\u5927\u5024\u30fb\u6700\u5c0f\u5024\u3092\u6c42\u3081\u308b\u3002<\/p>\n
\u9670\u95a2\u6570\u5b9a\u7406\u3088\u308a
\n$$ \\frac{dy}{dx} = – \\frac{ \\ \\ \\frac{\\partial f}{\\partial x} \\ \\ }{\\frac{\\partial f}{\\partial y}} = – \\frac{2 x}{2 y} = – \\frac{x}{y}$$<\/p>\n
\\(y(x)\\) \u304c\u6975\u5024\u3092\u3068\u308b\u306e\u306f \\(\\displaystyle \\frac{dy}{dx} = 0\\) \u3088\u308a \\(x = 0\\)\u3002\u3053\u308c\u3092 \\(f(x, y)\\) \u306b\u4ee3\u5165\u3057\u3066<\/p>\n
$$f(0, y) = y^2 -1 = 0, \\quad \\therefore\\ \\ y = \\pm 1$$<\/p>\n
\u3057\u305f\u304c\u3063\u3066\u6975\u5024\u306b\u306a\u308a\u305d\u3046\u306a\u70b9\u306f \\((x, y) = (0, 1), \\ (0, -1)\\)\u3002<\/p>\n
\u6700\u5927\u5024\u30fb\u6700\u5c0f\u5024\u306b\u306a\u308b\u304b\u3069\u3046\u304b\u3092\u8abf\u3079\u308b\u306b\u306f2\u968e\u5fae\u5206 \\(\\displaystyle \\frac{d^2 y}{dx^2}\\) \u3092\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\frac{d^2 y}{dx^2} &=& \\frac{d}{dx}\\left( -\\frac{x}{y}\\right) \\\\
\n&=& -\\frac{1}{y^2} \\left(1\\cdot y -x \\frac{dy}{dx} \\right) \\\\
\n&=& -\\frac{1}{y^2} \\left(y +\\frac{x^2}{y} \\right)\\\\
\n&=& \\left\\{
\n\\begin{array}{ll}
\n-1 < 0 & \\mbox{for}\\ (x, y) = (0, 1)\\\\
\n\\ \\ 1 > 0 & \\mbox{for}\\ (x, y) = (0, -1)
\n\\end{array}
\n\\right.
\n\\end{eqnarray}<\/p>\n
\u3057\u305f\u304c\u3063\u3066\uff0c\\((x,y) = (0, 1) \\) \u306f\u6700\u5927\u5024 \\(y = 1\\) \u3068\u306a\u308b\u70b9\u3067\u3042\u308a\uff0c\\( (x,y) = (0, -1)\\) \u306f\u6700\u5c0f\u5024 \\(y=-1\\) \u3068\u306a\u308b\u70b9\u3002<\/p>\n
\\(f(x, y) = x^2 + y^2 -1 = 0 \\) \u306f\u967d\u95a2\u6570\u3068\u3057\u3066\u8868\u3059\u3053\u3068\u3082\u53ef\u80fd\u3067\uff0c
\n$$x^2 + y^2 -1 = 0 \\ \\ \\Rightarrow\\ \\ y = \\pm \\sqrt{1-x^2}$$<\/p>\n
\\(y = +\\sqrt{1-x^2}\\) \u306e\u3068\u304d\u306f $$\\frac{dy}{dx} = \\frac{1}{2} (1-x^2)^{-\\frac{1}{2}} \\cdot(-2 x) = -\\frac{x}{y}$$<\/p>\n
\\(y = -\\sqrt{1-x^2}\\) \u306e\u3068\u304d\u306f $$\\frac{dy}{dx} = -\\frac{1}{2} (1-x^2)^{-\\frac{1}{2}} \\cdot(-2 x) = -\\frac{x}{y}$$<\/p>\n
\u3068\u306a\u308a\uff0c\u3069\u3061\u3089\u306e\u5834\u5408\u3082 $$\\frac{dy}{dx} = – \\frac{x}{y}$$<\/p>\n
\\(\\displaystyle \\frac{dy}{dx}\\) \u304c\u6c42\u307e\u308c\u3070\uff0c\u3042\u3068\u306f\u540c\u69d8\u306b…<\/p>\n
\u3053\u306e\u4f8b\u984c\u304c\u9670\u95a2\u6570\u5b9a\u7406\u306e\u4f8b\u984c\u3067\u3042\u308b\u3068\u3044\u3046\u5074\u9762\u3092\u96e2\u308c\u3066\uff0c\u5e7e\u4f55\u5b66\u7684\u306b\u8003\u5bdf\u3057\u3066\u307f\u308b\u3068\uff0c$f(x, y) = x^2 + y^2 -1 = 0$ \u3068\u3044\u3046\u62d8\u675f\u6761\u4ef6\u306f\uff0c\u70b9 $(x, y)$ \u304c $x^2 + y^2 = 1$ \u3064\u307e\u308a\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u534a\u5f84 $1$ \u306e\u5186\u4e0a\u306b\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3002<\/p>\n
<\/p>\n
\u3053\u306e\u62d8\u675f\u6761\u4ef6\u306e\u3082\u3068\u3067 $y$ \u306e\u6700\u5927\u5024\u30fb\u6700\u5c0f\u5024\u3092\u6c42\u3081\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u3053\u306e\u5186\u4e0a\u3067 $y$ \u306e\u6700\u5927\u5024\u3068\u6700\u5c0f\u5024\u3092\u6c42\u3081\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u304b\u3089\uff0c\u56f3\u304b\u3089\u660e\u3089\u304b\u306b $y$ \u306e\u6700\u5927\u5024\u306f $1$\uff0c\u6700\u5c0f\u5024\u306f $-1$\u00a0 \u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n
\\(f(x, y) = x^2 + y^2 -1 = 0\\) \u3092\u6e80\u305f\u3059\u9670\u95a2\u6570\u3092 \\(y=y(x)\\) \u3068\u3059\u308b\u3068\u304d\uff0c\u95a2\u6570 \\(g(x) = x + y(x)\\) \u306e\u6700\u5927\u5024\u30fb\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088\u3002<\/p>\n
\u3059\u3067\u306b $\\displaystyle \\frac{dy}{dx} = – \\frac{x}{y}$ \u307e\u3067\u306f\u6c42\u3081\u3066\u3044\u308b\u306e\u3067\uff0c\u3042\u3068\u306f $g(x)$ \u306e\u6975\u5024\u3092\u63a2\u3059\u305f\u3081\u306b $\\displaystyle \\frac{dg}{dx}$ \u3092\u6c42\u3081\u3066\u307f\u308c\u3070\u3088\u3044\u3002<\/p>\n
\\begin{eqnarray}
\n\\frac{d}{dx} g(x) &=& \\frac{d}{dx} (x + y) \\\\
\n&=& 1 + \\frac{dy}{dx} \\\\
\n&=& 1 -\\frac{x}{y} = \\frac{y -x}{y}
\n\\end{eqnarray}<\/p>\n
\u6975\u5024\u3068\u306a\u308a\u305d\u3046\u306a\u70b9\u306f $\\displaystyle \\frac{dg}{dx} = 0$ \u3088\u308a $y = x$ \u3092\u6e80\u305f\u3059\u3053\u3068\u306b\u306a\u308b\u3002\u3042\u3068\u306f…<\/p>\n
\u95a2\u6570 \\(g(x) = x + y(x)\\) \u306e\u5024\u3092 $d$ \u3068\u304a\u3051\u3070\uff0c<\/p>\n
$$x + y = d, \\quad \\therefore\\ \\ y = -x + d$$<\/p>\n
\u3064\u307e\u308a\uff0c\u3053\u306e\u7df4\u7fd2\u554f\u984c\u3092\u5e7e\u4f55\u5b66\u7684\u306b\u8003\u5bdf\u3059\u308b\u3068\uff0c\u5186 $x^2 + y^2 = 1$ \u3068\u5171\u901a\u70b9\u3092\u3082\u3064\uff08\u3064\u307e\u308a\uff0c\u4ea4\u308f\u308b\u3042\u308b\u3044\u306f\u63a5\u3059\u308b\uff09\u76f4\u7dda $y = -x + d$ \u306e\u3046\u3061\uff0c\u5207\u7247 $d$ \u306e\u5024\u304c\u6700\u5927\u30fb\u6700\u5c0f\u306b\u306a\u308b\u306e\u306f\uff1f\u3068\u3044\u3046\u554f\u984c\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u3053\u308c\u306a\u3089\uff0c\u9670\u95a2\u6570\u5b9a\u7406\u3084\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\uff08\u307e\u3060\u6559\u3048\u3066\u306a\u3044\u3051\u3069\uff09\u306a\u3093\u304b\u4f7f\u308f\u306a\u304f\u3066\u3082\uff0c\u56f3\u3092\u66f8\u3044\u3066\u6c42\u3081\u3089\u308c\u308b\u3088\u306d\u3002<\/p>\n