{"id":2075,"date":"2022-02-22T12:11:35","date_gmt":"2022-02-22T03:11:35","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2075"},"modified":"2024-04-17T10:20:55","modified_gmt":"2024-04-17T01:20:55","slug":"%e5%be%ae%e5%88%86%e6%b3%95%e3%81%ae%e5%85%ac%e5%bc%8f","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e5%be%ae%e5%88%86%e6%b3%95%e3%81%ae%e5%85%ac%e5%bc%8f\/","title":{"rendered":"\u5fae\u5206\u6cd5\u306e\u516c\u5f0f"},"content":{"rendered":"

\u4e3b\u306a\u516c\u5f0f<\/h3>\n
    \n
  1. \\(\\displaystyle \\left\\{ f(x) \\pm g(x) \\right\\}’ = f'(x) \\pm g'(x) \\)<\/li>\n
  2. \\(\\displaystyle \\left\\{ c f(x) \\right\\}’ = c f'(x) \\)\u00a0 \uff08\\(c\\) \u306f\u5b9a\u6570\uff09<\/li>\n
  3. \\(\\displaystyle \\left\\{ f(x)\\, g(x) \\right\\}’ = f'(x)\\,g(x) + f(x)\\,g'(x) \\) \uff08\u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u30eb\u30fc\u30eb\uff09<\/li>\n
  4. \\(\\displaystyle \\left\\{ \\frac{1}{g(x)} \\right\\}’ = -\\frac{g'(x)}{\\left\\{g(x)\\right\\}^2} \\)<\/li>\n
  5. \\(\\displaystyle \\left\\{ \\frac{f(x)}{g(x)} \\right\\}’ = \\frac{f'(x)\\,g(x) -f(x)\\, g'(x)}{\\left\\{g(x)\\right\\}^2} \\)<\/li>\n<\/ol>\n

    <\/p>\n

    \u8a3c\u660e<\/h4>\n

    1. \u304a\u3088\u3073 2. \u306b\u3064\u3044\u3066\u306f\u7701\u7565\u3002<\/p>\n

    3. \u306e\u8a3c\u660e\uff1a<\/p>\n

    \\begin{eqnarray}
    \n\\left\\{ f(x)\\, g(x) \\right\\}’ &=& \\lim_{h \\rightarrow 0} \\frac{f(x+h)\\,g(x+h) -f(x)\\,g(x)}{h} \\\\
    \n&=& \\lim_{h \\rightarrow 0} \\frac{f(x+h)\\,g(x+h) -f(x)\\,g(x+h) + f(x)\\,g(x+h) -f(x)\\,g(x)}{h}\\\\
    \n&=& \\lim_{h \\rightarrow 0} \\frac{f(x+h) -f(x)}{h}\\,g(x+h) + \\lim_{h \\rightarrow 0}\\frac{f(x) \\{g(x+h) -g(x)\\}}{h}\\\\
    \n&=&\\lim_{h \\rightarrow 0} \\frac{f(x+h) -f(x)}{h}\\,g(x) +f(x) \\lim_{h \\rightarrow 0}\\frac{g(x+h) -g(x)}{h}\\\\
    \n&=& f'(x)\\,g(x) + f(x)\\,g'(x)
    \n\\end{eqnarray}<\/p>\n

    4. \u306e\u8a3c\u660e\uff1a<\/p>\n

    \\begin{eqnarray}
    \n\\left\\{ \\frac{1}{g(x)} \\right\\}’ &=& \\lim_{h \\rightarrow 0} \\frac{1}{h} \\left\\{\\frac{1}{g(x+h)} -\\frac{1}{g(x)}\\right\\}\\\\
    \n&=& \\lim_{h \\rightarrow 0} \\frac{1}{h} \\frac{g(x) -g(x+h)}{g(x+h)\\,g(x)} \\\\
    \n&=& -\\frac{1}{\\{g(x)\\}^2}\\,\\lim_{h \\rightarrow 0} \\frac{g(x+h) -g(x)}{h} \\\\
    \n&=& -\\frac{g'(x)}{\\{g(x)\\}^2}
    \n\\end{eqnarray}<\/p>\n

    5. \u306b\u3064\u3044\u3066\u306f\uff0c<\/p>\n

    $$
    \n\\left\\{ \\frac{f(x)}{g(x)} \\right\\}’ = \\left\\{ f(x) \\cdot \\frac{1}{g(x)} \\right\\}’ = f'(x)\\cdot \\frac{1}{g(x)} + f(x) \\cdot\\left\\{ \\frac{1}{g(x)} \\right\\}’$$ \u3068\u3059\u308c\u3070\uff0c3. \u304a\u3088\u3073 4. \u306e\u516c\u5f0f\u304b\u3089\u7c21\u5358\u306b\u8a3c\u660e\u3067\u304d\u308b\u3002<\/p>\n

    \u5408\u6210\u95a2\u6570\u306e\u5fae\u5206\u6cd5<\/h3>\n

    \\( y = g(u), \\ u = f(x) \\) \u3064\u307e\u308a\uff0c\\( y\\) \u304c \\(u\\) \u3092\u901a\u3057\u3066 \\(x\\) \u306e\u95a2\u6570\u3067\u3042\u308b\u3068\u304d\uff0c<\/p>\n

    $$ \\frac{dy}{dx} = \\frac{dy}{du}\\,\\frac{du}{dx} $$<\/p>\n

    \u9ad8\u6821\u306e\u6570\u5b66\u3067\u306f \\(\\displaystyle \\frac{dy}{dx}\\) \u306f\u5206\u6570\u3067\u306f\u306a\u3044\uff01\u3068\u7fd2\u3063\u305f\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u5927\u5b66\u3067\u306f\u5206\u6570\u307f\u305f\u3044\u306a\u3082\u306e\u3067\u3042\u308b\uff0c\u3068\u3057\u3061\u3083\u3046\u306e\u3067\uff0c\u4e0a\u8a18\u306e\u8a3c\u660e\u306f\u5206\u6bcd\u5206\u5b50\u306e \\(du\\) \u304c\u7d04\u5206\u3055\u308c\u308b\u304b\u3089\u660e\u3089\u304b\u3067\u3042\u308d\u3046\u3002<\/p>\n

    \u4f8b\uff1a<\/p>\n

    \\( y =\u00a0 (a x^2 + b x + c)^5\u00a0 \\) \u306e\u5c0e\u95a2\u6570 \\( y’\\) \u306e\u8a08\u7b97\u306f\uff0c\\( u \\equiv a x^2 + b x + c \\) \u3068\u3057\u3066 \\( y = u^5\\) \u3060\u304b\u3089\uff0c
    \n$$ y’ = \\frac{dy}{du} \\frac{du}{dx} = 5 u^4 (2 a x + b) = 5 (2 a x + b)(a x^2 + b x + c)^4$$<\/p>\n

    \u9006\u95a2\u6570\u306e\u5fae\u5206\u6cd5<\/h3>\n

    \\( y = f(x) \\) \u306e\u9006\u95a2\u6570 \\(x = f^{-1} (y) \\) \u306e\u5fae\u5206\u306f\uff0c
    \n$$ \\frac{dx}{dy} = \\frac{1}{\\frac{dy}{dx}} $$<\/p>\n

    \u8a3c\u660e\uff1a<\/p>\n

    \u5408\u6210\u95a2\u6570\u306e\u5fae\u5206\u3088\u308a
    \n$$ \\frac{dx}{dx} = \\frac{dx}{dy}\\,\\frac{dy}{dx} = 1, \\ \\ \\therefore \\ \\frac{dx}{dy} = \\frac{1}{\\frac{dy}{dx}}$$<\/p>\n

    \u4f8b\uff1a<\/p>\n

    \\( y = a x + b \\) (\\( a\\neq 0\\) ) \u306e\u9006\u95a2\u6570\u306f\uff0c
    \n$$ y -b = a x, \\ \\ \\therefore \\ \\ x = \\frac{y -b}{a} $$
    \n$$ \\frac{dy}{dx} = \\frac{d}{dx} (a x + b) = a, \\quad \\frac{dx}{dy} = \\frac{d}{dx} \\left( \\frac{y -b}{a}\\right) = \\frac{1}{a}$$ \u3060\u304b\u3089\uff0c\u78ba\u304b\u306b
    \n$$\\frac{dx}{dy} = \\frac{1}{\\frac{dy}{dx}}$$\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n

    \u30d1\u30e9\u30e1\u30fc\u30bf\u8868\u793a\u306e\u95a2\u6570\u306e\u5fae\u5206\u6cd5<\/h3>\n

    \\(x = f(t), \\ y = g(t)\\) \u306f\u5909\u6570\uff08\u30d1\u30e9\u30e1\u30fc\u30bf\uff09\\(t\\) \u306e\u5024\u3092\u6c7a\u3081\u308b\u3068 \\(x\\)\u00a0 \u3068 \\(y\\) \u306e\u5024\u3082\u4e00\u610f\u306b\u6c7a\u307e\u308b\u3002\u8a00\u3044\u63db\u3048\u308b\u3068\uff0c\\(y\\) \u306f\u30d1\u30e9\u30e1\u30fc\u30bf \\(t\\) \u3092\u901a\u3057\u3066 \\(x\\) \u306e\u95a2\u6570\u3068\u307f\u306a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u306e\u3068\u304d\uff0c
    \n$$ \\frac{dy}{dx} = \\frac{dy}{dt}\\,\\frac{dt}{dx} = \\frac{\\frac{dy}{dt}}{\\frac{dx}{dt}} = \\frac{\\dot{y}}{\\dot{x}} $$<\/p>\n","protected":false},"excerpt":{"rendered":"

    \u4e3b\u306a\u516c\u5f0f <\/p>

    \u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n

  6. \\(\\displaystyle \\left\\{ f(x) \\pm g(x) \\right\\}’ = f'(x) \\pm g'(x) \\)<\/li>\n
  7. \\(\\displaystyle \\left\\{ c f(x) \\right\\}’ = c f'(x) \\)\u00a0 \uff08\\(c\\) \u306f\u5b9a\u6570\uff09<\/li>\n
  8. \\(\\displaystyle \\left\\{ f(x)\\, g(x) \\right\\}’ = f'(x)\\,g(x) + f(x)\\,g'(x) \\) \uff08\u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u30eb\u30fc\u30eb\uff09<\/li>\n
  9. \\(\\displaystyle \\left\\{ \\frac{1}{g(x)} \\right\\}’ = -\\frac{g'(x)}{\\left\\{g(x)\\right\\}^2} \\)<\/li>\n
  10. \\(\\displaystyle \\left\\{ \\frac{f(x)}{g(x)} \\right\\}’ = \\frac{f'(x)\\,g(x) -f(x)\\, g'(x)}{\\left\\{g(x)\\right\\}^2} \\)<\/li>\n","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2075","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2075","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=2075"}],"version-history":[{"count":4,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2075\/revisions"}],"predecessor-version":[{"id":8413,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2075\/revisions\/8413"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2068"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2075"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}