{"id":14429,"date":"2025-12-24T17:20:52","date_gmt":"2025-12-24T08:20:52","guid":{"rendered":"https:\/\/math-travel.com\/?p=14429"},"modified":"2026-02-11T17:05:42","modified_gmt":"2026-02-11T08:05:42","slug":"xn-differential","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-2\/xn-differential\/","title":{"rendered":"x^n\u306e\u5fae\u5206\u516c\u5f0f\u3068\u5c0e\u304d\u65b9\uff01\u591a\u9805\u5f0f\u306e\u5fae\u5206\u3092\u4e00\u77ac\u3067\u89e3\u304f\u305f\u3081\u306e\u57fa\u672c\u30eb\u30fc\u30eb"},"content":{"rendered":"\n<div class=\"wp-block-group has-border -border02 is-style-bg_grid\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div id=\"kv\">\n        <p>\u5165\u529b\u3057\u305f\u95a2\u6570\u3092\u5fae\u5206\u3057\u307e\u3059\u3002<br>\n          \u95a2\u6570\u3092\u5165\u529b\u5f8c\u306b\u7b97\u51fa\u30dc\u30bf\u30f3\u30af\u30ea\u30c3\u30af\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n      <\/div>\n      <div>\n        <input class=\"input-degree\" type=\"number\" onchange=\"changeDegree(this)\">\n        <span>\u6b21\u5f0f<\/span>\n      <\/div>\n      <div class=\"input-form\">\n        <p id=\"input-formula\"><\/p>\n        <div id=\"input-form-inner\"><\/div>\n      <\/div>\n      <div class=\"output\">\n        <p>\u95a2\u6570\uff1a<span id=\"origin-formula\"><\/span>\u3092\u5fae\u5206\u3059\u308b\u3068\u3001<span class=\"sp\"><br><\/span><span id=\"output-formula\"><\/span>\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n      <\/div>\n      <div class=\"btn\">\n        <input type=\"button\" value=\"\u7b97\u51fa\" id=\"cal-btn\">\n        <input type=\"button\" value=\"\u30ea\u30bb\u30c3\u30c8\" id=\"reset-btn\">\n      <\/div>\n<\/div><\/div>\n\n\n\n<p>\u4eca\u56de\u306f\u6570\u5b66\u2161\u306e\u5fae\u5206\u7a4d\u5206\u304b\u3089<span class=\"huto\">\u300c\\(x^{n}\\)\u306e\u5fae\u5206\u300d<\/span>\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"is-style-bg_stripe\"><span class=\"huto\">\u300c\u5fae\u5206\u306e\u3084\u308a\u65b9\u304c\u5206\u304b\u3089\u306a\u3044\u300d<\/span><br><span class=\"huto\">\u300c\u5fae\u5206\u306e\u516c\u5f0f\u3092\u307e\u3068\u3081\u3066\u6b32\u3057\u3044\u300d<\/span><\/p>\n\n\n<div class=\"swell-block-balloon\"><div class=\"c-balloon -bln-left\" data-col=\"gray\"><div class=\"c-balloon__icon -circle\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/math-travel.jp\/wp-content\/uploads\/2024\/05\/math-travel-hukidashi5-150x150.webp\" alt=\"\" class=\"c-balloon__iconImg\" width=\"80px\" height=\"80px\"><span class=\"c-balloon__iconName\">\u9ad8\u6821\u751f<\/span><\/div><div class=\"c-balloon__body -speaking -border-on\"><div class=\"c-balloon__text\">\n<p>\u5fae\u5206\u6cd5\u3092\u7fd2\u3044\u59cb\u3081\u305f\u3070\u304b\u308a\u3067\u2026<\/p>\n<span class=\"c-balloon__shapes\"><span class=\"c-balloon__before\"><\/span><span class=\"c-balloon__after\"><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n<p>\u5fae\u5206\u3092\u6d3b\u7528\u3059\u308b\u3068<span class=\"hutoaka\">\u30b0\u30e9\u30d5\u306e\u5f62\u3084\u76f4\u7dda\u306e\u50be\u304d\u306a\u3069\u3092\u6c42\u3081\u308b<\/span>\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u4eca\u5f8c\u3001\u5fae\u5206\u3092\u4f7f\u3044\u3053\u306a\u3057\u3066\u3044\u304f\u305f\u3081\u306b\u3082\u3001<span class=\"yellow-hoso\">\u5fae\u5206\u306e\u57fa\u672c\u516c\u5f0f\u306e\u7406\u89e3\u306f\u5fc5\u9808<\/span>\u3067\u3059\u3002<\/p>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u5fae\u5206\u306e\u57fa\u672c\u516c\u5f0f<\/span><\/div><div class=\"cap_box_content\">\n<p>\\[(x^{n})^ \\prime =n x^{n-1}\\]<\/p>\n<\/div><\/div>\n\n\n\n<p>\u6307\u6570\u306e\\(n\\)\u304c\u4fc2\u6570\u3068\u3057\u3066\u524d\u306b\u964d\u308a\u3066\u304d\u3066\u3001\u6307\u6570\u306e\u5024\u306f1\u5c0f\u3055\u304f\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u306e\u8a18\u4e8b\u3067\u306f\u5fae\u5206\u306e\u8d85\u57fa\u672c\u3068\u306a\u308b\u516c\u5f0f\u3092\u7d39\u4ecb\u3057\u3001<span class=\"^yellow-hoso\">\u5b9f\u969b\u306b\u5fae\u5206\u306e\u3084\u308a\u65b9\u3092\u89e3\u8aac<\/span>\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p><span class=\"hutoaka\">\u3068\u3066\u3082\u91cd\u8981\u306a\u3068\u3053\u308d<\/span>\u306a\u306e\u3067\u3057\u3063\u304b\u308a\u7406\u89e3\u3067\u304d\u308b\u3088\u3046\u306b\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\uff01<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"bibun\">\\(x^{n}\\)\u3079\u304d\u95a2\u6570\u306e\u5fae\u5206<\/h2>\n\n\n\n<p>\\(x^{n}\\)\u3092\u5fae\u5206\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u5fae\u5206\u306e\u57fa\u672c\u516c\u5f0f<\/span><\/div><div class=\"cap_box_content\">\n<p>\\[(x^{n})^ \\prime =n x^{n-1}\\]<\/p>\n<\/div><\/div>\n\n\n\n<p>\u4f8b\u3068\u3057\u3066\\(f(x)=x^{3}\\)\u3068\u3044\u3046\u5f0f\u3092\u5fae\u5206\u3059\u308b\u3068<\/p>\n\n\n\n<p>\\begin{eqnarray}<br>f^{\\prime} (x)&amp;=&amp;(x^{3})^{\\prime}\\\\<br>&amp;=&amp;3x^{2}<br>\\end{eqnarray}<\/p>\n\n\n\n<p>\u3053\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u5408\u308f\u305b\u3066\u3001\u5fae\u5206\u3068\u306f\u4f55\u304b\u3092\u8aac\u660e\u3057\u3066\u304a\u304f\u3068\u3001<\/p>\n\n\n\n<p>\u95a2\u6570\\(f(x)\\)\u304b\u3089\u5c0e\u95a2\u6570\\(f^{\\prime} (x)\\)\u3092\u6c42\u3081\u308b\u3053\u3068\u3092\u3001\\(f(x)\\)\u3092\\(x\\)\u3067\u5fae\u5206\u3059\u308b\u307e\u305f\u306f\u5358\u306b\u5fae\u5206\u3059\u308b\u3068\u3044\u3046\u3002<\/p>\n\n\n\n<p>\u3064\u307e\u308a\u3001<\/p>\n\n\n\n<p class=\"is-style-bg_stripe\">\u5fae\u5206\u3059\u308b\uff1d\u5c0e\u95a2\u6570\u3092\u6c42\u3081\u308b<\/p>\n\n\n\n<p>\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"syoumei\">\\(x^{n}\\)\u306e\u5fae\u5206\u516c\u5f0f\u3000\u8a3c\u660e<\/h2>\n\n\n\n<p>\u3053\u3053\u304b\u3089\u306f\u3001\\(x^{n}\\)\u306e\u5fae\u5206\u306e\u516c\u5f0f\u30fb\u8a3c\u660e\u3092\u7d39\u4ecb\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p><span class=\"yellow-hoso\">\\(x^{n}\\)\u306e\u5fae\u5206\u306e\u516c\u5f0f\u306f\u3053\u308c\u304b\u3089\u5fae\u5206\u306e\u8a08\u7b97\u3092\u3059\u308b\u3068\u304d\u306b\u975e\u5e38\u306b\u91cd\u8981\u306a\u516c\u5f0f\u3067\u3059\u3002<\/span><\/p>\n\n\n\n<p>\u3057\u3063\u304b\u308a\u899a\u3048\u3066\u4f7f\u3048\u308b\u3088\u3046\u306b\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\uff01<\/p>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u5fae\u5206\u306e\u57fa\u672c\u516c\u5f0f<\/span><\/div><div class=\"cap_box_content\">\n<p>\\[(x^{n})^ \\prime =n x^{n-1}\\]<\/p>\n<\/div><\/div>\n\n\n\n<p>\u3067\u306f\u5b9f\u969b\u306b\u3053\u306e\u516c\u5f0f\u306e\u8a3c\u660e\u3092\u7d39\u4ecb\u3057\u3066\u3044\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3010\u8a3c\u660e\u3011<br>\u5c0e\u95a2\u6570\u306e\u5b9a\u7fa9\u3088\u308a\u3001<\/p>\n\n\n\n<p>\\[x^{n}=\\lim_{h \\to 0} {\\frac{{(x+h)}^n-x^n}{h}}\\]<\/p>\n\n\n\n<p>\u4e8c\u9805\u5b9a\u7406\u3088\u308a\u3001<\/p>\n\n\n\n<p>\\((x+h)^{n}\\)<br>\\(=nC_{0} x^{n}+nC_{1} x^{n-1} h+\\cdots+nC_{n} h^{n}\\)<br>\\(=x^{n}+{nx}^{n-1}h+\\cdots+h^{n}\\)<\/p>\n\n\n\n<p>\u3088\u3063\u3066\u3001<\/p>\n\n\n\n<p>\\[(x+h)^{n}-x^{n}={nx}^{n-1}h+\\cdots+h^{n}\\]<\/p>\n\n\n\n<p>\u4e21\u8fba\u30920\u3067\u306a\u3044\u6570\\(h\\)\u3067\u5272\u308b\u3068<\/p>\n\n\n\n<p>\\[\\displaystyle \\frac{(x+h)^{n}-x^{n}}{h}={nx}^{n-1}+\\cdots+h^{n-1}\\]<\/p>\n\n\n\n<p>\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p>\\[\\displaystyle \\lim_{h \\to 0} {\\frac{(x+h)^{n}-x^{n}}{h}}={nx}^{n-1}\\]<\/p>\n\n\n\n<p>\u3059\u306a\u308f\u3061<\/p>\n\n\n\n<p><span class=\"hutoaka\">\\[(x^{n})^{\\prime}={nx}^{n-1}\\]<\/span><\/p>\n\n\n<div class=\"swell-block-balloon\"><div class=\"c-balloon -bln-left\" data-col=\"gray\"><div class=\"c-balloon__icon -circle\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/math-travel.jp\/wp-content\/uploads\/2024\/05\/math-travel-hukidashi7-150x150.webp\" alt=\"\" class=\"c-balloon__iconImg\" width=\"80px\" height=\"80px\"><span class=\"c-balloon__iconName\">\u9ad8\u6821\u751f<\/span><\/div><div class=\"c-balloon__body -speaking -border-on\"><div class=\"c-balloon__text\">\n<p>\u4e8c\u9805\u5b9a\u7406\u3092\u7528\u3044\u3066\u8a3c\u660e\u3067\u304d\u308b\u3093\u3067\u3059\u306d\uff01<\/p>\n<span class=\"c-balloon__shapes\"><span class=\"c-balloon__before\"><\/span><span class=\"c-balloon__after\"><\/span><\/span><\/div><\/div><\/div><\/div>\n\n<div class=\"swell-block-balloon\"><div class=\"c-balloon -bln-right\" data-col=\"gray\"><div class=\"c-balloon__icon -circle\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/math-travel.jp\/wp-content\/uploads\/2024\/05\/math-travel-hukidashi1-150x150.webp\" alt=\"\" class=\"c-balloon__iconImg\" width=\"80px\" height=\"80px\"><span class=\"c-balloon__iconName\">\u30b7\u30fc\u30bf<\/span><\/div><div class=\"c-balloon__body -speaking -border-on\"><div class=\"c-balloon__text\">\n<p>\u6570\u5b66\u306f\u5206\u91ce\u3092\u8d85\u3048\u3066\u7e4b\u304c\u3063\u3066\u3044\u308b\u3093\u3060\u306d<\/p>\n<span class=\"c-balloon__shapes\"><span class=\"c-balloon__before\"><\/span><span class=\"c-balloon__after\"><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"mondai\">\\(x^{n}\\)\u306e\u5fae\u5206\u300a\u7df4\u7fd2\u554f\u984c\u300b<\/h2>\n\n\n\n<p>\u3053\u3053\u304b\u3089\u306f\u3001\u5b9f\u969b\u306b\\(x^{n}\\)\u306e\u5fae\u5206\u306e\u7df4\u7fd2\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046\uff01<\/p>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u7df4\u7fd2\u554f\u984c<\/span><\/div><div class=\"cap_box_content\">\n<p>\u6b21\u306e\u95a2\u6570\u3092\u5fae\u5206\u3057\u306a\u3055\u3044\u3002<\/p>\n\n\n\n<p>(1) \\(f(x)=x^{4}\\)<\/p>\n\n\n\n<p>(2) \\(f(x)=x^{5}\\)<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"swell-block-accordion is-style-default\">\n<details class=\"swell-block-accordion__item\" data-swl-acc=\"wrapper\"><summary class=\"swell-block-accordion__title\" data-swl-acc=\"header\"><span class=\"swell-block-accordion__label\"><\/span><span class=\"swell-block-accordion__icon c-switchIconBtn\" data-swl-acc=\"icon\" aria-hidden=\"true\" data-opened=\"false\"><i class=\"__icon--closed icon-chevron-down\"><\/i><i class=\"__icon--opened icon-chevron-up\"><\/i><\/span><\/summary><div class=\"swell-block-accordion__body\" data-swl-acc=\"body\">\n\n<\/div><\/details>\n<\/div>\n\n\n\n<div class=\"swell-block-accordion\">\n<details class=\"swell-block-accordion__item\" data-swl-acc=\"wrapper\"><summary class=\"swell-block-accordion__title\" data-swl-acc=\"header\"><span class=\"swell-block-accordion__label\">\u2003\u89e3\u7b54\u3092\u78ba\u8a8d\u3059\u308b<\/span><span class=\"swell-block-accordion__icon c-switchIconBtn\" data-swl-acc=\"icon\" aria-hidden=\"true\" data-opened=\"false\"><i class=\"__icon--closed icon-caret-down\"><\/i><i class=\"__icon--opened icon-caret-up\"><\/i><\/span><\/summary><div class=\"swell-block-accordion__body\" data-swl-acc=\"body\">\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u89e3\u7b54<\/span><\/div><div class=\"cap_box_content\">\n<p>(1) \\(f^{\\prime}(x)=4x^{3}\\)<\/p>\n\n\n\n<p>(2) \\(f^{\\prime}(x)=5x^{4}\\)<\/p>\n<\/div><\/div>\n<\/div><\/details>\n<\/div>\n\n\n\n<p>\uff08\uff11\uff09\uff08\uff12\uff09\u3092\u307e\u3068\u3081\u3066\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u306e\u554f\u984c\u306f\u4e21\u65b9\u3068\u3082\u516c\u5f0f\u306b\u5f53\u3066\u306f\u3081\u3066\u8a08\u7b97\u3059\u308c\u3070\u3067\u304d\u308b\u306e\u3067\u3059\u304c\u3001\u899a\u3048\u3084\u3059\u3044\u8003\u3048\u65b9\u3092\u7d39\u4ecb\u3057\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002\u56f3\u3092\u8f09\u305b\u3066\u304a\u304d\u307e\u3059\u306e\u3067\u3001\u305d\u308c\u3092\u898b\u306a\u304c\u3089\u8003\u3048\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<div class=\"st-memobox2\">\n<p class=\"st-memobox-title\">\u5fae\u5206\u306e\u624b\u9806<\/p>\n<ol>\n<li>\u5fae\u5206\u3057\u305f\u95a2\u6570\u306e\u4fc2\u6570\u3092\u8003\u3048\u308b<\/li>\n<li>\u5fae\u5206\u3057\u305f\u95a2\u6570\u306e\u6307\u6570\u3092\u8003\u3048\u308b<\/li>\n<\/ol>\n<\/div>\n\n\n\n<p>\uff08\uff11\uff09\u306e\u554f\u984c\u3092\u4f7f\u3063\u3066\u8aac\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\\(f(x)=x^{4}\\) \u3092\u5fae\u5206\u3057\u305f\u3068\u304d\u3001x\u306e\u4fc2\u6570\uff11\u3068\u6307\u65704\u3092\u304b\u3051\u305f\u5024\u304c\u5fae\u5206\u3057\u305f\u95a2\u6570\u306e\u4fc2\u6570\uff14\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" width=\"661\" height=\"248\" src=\"https:\/\/math-travel.jp\/wp-content\/uploads\/2023\/11\/xn\u5fae\u5206\u306e\u3084\u308a\u65b9-e1699244032170.png\" alt=\"xn\u5fae\u5206\u306e\u3084\u308a\u65b9\" class=\"wp-image-14607\" srcset=\"https:\/\/math-travel.jp\/wp-content\/uploads\/2023\/11\/xn\u5fae\u5206\u306e\u3084\u308a\u65b9-e1699244032170.png 661w, https:\/\/math-travel.jp\/wp-content\/uploads\/2023\/11\/xn\u5fae\u5206\u306e\u3084\u308a\u65b9-e1699244032170-300x113.png 300w\" sizes=\"(max-width: 661px) 100vw, 661px\" \/><\/figure>\n<\/div>\n\n\n<p>\u6b21\u306b\u624b\u98062\u3067\u3059\u3002\u5143\u306e\u95a2\u6570\u306e\u6307\u65704\u304b\u30891\u3092\u5f15\u3044\u305f\u65703\u304c\u5fae\u5206\u3057\u305f\u95a2\u6570\u306e\u6307\u6570\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" width=\"641\" height=\"249\" src=\"https:\/\/math-travel.jp\/wp-content\/uploads\/2023\/11\/xn\u5fae\u5206\u306e\u3084\u308a\u65b9\u2460-e1699244054968.png\" alt=\"xn\u5fae\u5206\u306e\u3084\u308a\u65b9\u2460\" class=\"wp-image-14608\" srcset=\"https:\/\/math-travel.jp\/wp-content\/uploads\/2023\/11\/xn\u5fae\u5206\u306e\u3084\u308a\u65b9\u2460-e1699244054968.png 641w, https:\/\/math-travel.jp\/wp-content\/uploads\/2023\/11\/xn\u5fae\u5206\u306e\u3084\u308a\u65b9\u2460-e1699244054968-300x117.png 300w\" sizes=\"(max-width: 641px) 100vw, 641px\" \/><\/figure>\n<\/div>\n\n\n<p>\u5fae\u5206\u306f\u4fc2\u6570\u3068\u6307\u6570\u3001\u3053\u306e2\u3064\u3092\u8003\u3048\u308c\u3070\u7d42\u308f\u308a\u3067\u3059\u3002\u610f\u5916\u3068\u7c21\u5358\u3067\u3059\u3088\u306d\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"koushiki\">\u5fae\u5206\u306e\u91cd\u8981\u516c\u5f0f<\/h2>\n\n\n\n<p>\u4eca\u56de\u306f\u5fae\u5206\u306e\u57fa\u672c\u3068\u306a\u308b\\(x^{n}\\)\u306e\u5fae\u5206\u3092\u4e2d\u5fc3\u306b\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059\u304c\u3001<span class=\"yellow-hoso\">\u5fae\u5206\u306b\u306f\u5fc5\u305a\u899a\u3048\u3066\u304a\u304d\u305f\u3044\u516c\u5f0f\u304c\u3044\u304f\u3064\u304b\u3042\u308a\u307e\u3059\u3002<\/span><\/p>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u5fae\u5206\u306e\u91cd\u8981\u516c\u5f0f<\/span><\/div><div class=\"cap_box_content\">\n<ul class=\"wp-block-list\">\n<li><a href=\"#koushiki1\">\u5b9a\u6570\u500d\u306e\u5fae\u5206<\/a><\/li>\n\n\n\n<li><a href=\"#koushiki2\">\u548c\u3068\u5dee\u306e\u5fae\u5206<\/a><\/li>\n\n\n\n<li><a href=\"#koushiki3\">\u5b9a\u6570\u306e\u5fae\u5206<\/a><\/li>\n\n\n\n<li><a href=\"#koushiki4\">\u7a4d\u306e\u5fae\u5206<\/a><\/li>\n\n\n\n<li><a href=\"#koushiki5\">\u5546\u306e\u5fae\u5206<\/a><\/li>\n<\/ul>\n<\/div><\/div>\n\n\n<div class=\"swell-block-balloon\"><div class=\"c-balloon -bln-right\" data-col=\"gray\"><div class=\"c-balloon__icon -circle\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/math-travel.jp\/wp-content\/uploads\/2024\/05\/math-travel-hukidashi1-150x150.webp\" alt=\"\" class=\"c-balloon__iconImg\" width=\"80px\" height=\"80px\"><span class=\"c-balloon__iconName\">\u30b7\u30fc\u30bf<\/span><\/div><div class=\"c-balloon__body -speaking -border-on\"><div class=\"c-balloon__text\">\n<p>\u3069\u308c\u3082\u5fc5\u9808\u3060\u304b\u3089\u5fc5\u305a\u899a\u3048\u3088\u3046\uff01<\/p>\n<span class=\"c-balloon__shapes\"><span class=\"c-balloon__before\"><\/span><span class=\"c-balloon__after\"><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n<h3 class=\"wp-block-heading\" id=\"koushiki1\">\u5b9a\u6570\u500d\u306e\u5fae\u5206<\/h3>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u5b9a\u6570\u500d\u306e\u5fae\u5206<\/span><\/div><div class=\"cap_box_content\">\n<p>\\(y=k f(x)\\)\u3092\u5fae\u5206\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>\\[y^{\\prime}=k f^{\\prime}(x)\\]<\/p>\n<\/div><\/div>\n\n\n\n<p>\u3064\u307e\u308a\u3001\u5b9a\u6570\u90e8\u5206\u306f\u5909\u308f\u3089\u305a\\(x\\)\u3092\u6301\u3064\u90e8\u5206\u306e\u307f\u304c\u5fae\u5206\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u4f8b\u984c<\/span><\/div><div class=\"cap_box_content\">\n<p>\\(y=3{x}^{2}\\)\u3092\u5fae\u5206\u3057\u306a\u3055\u3044\u3002<\/p>\n<\/div><\/div>\n\n\n\n<p>\\(x^{n}\\)\u3068\u540c\u3058\u3088\u3046\u306b\u5fae\u5206\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u6307\u6570\u3092\u524d\u306b\u6301\u3063\u3066\u304d\u3066\u3001\u6307\u6570\u306e\u6570\u5b57\u306f1\u4e0b\u3052\u308b\u3068<\/p>\n\n\n\n<p><span class=\"huto\">\\[y^{\\prime}=6x\\]<\/span><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"koushiki2\">\u548c\u3068\u5dee\u306e\u5fae\u5206<\/h3>\n\n\n\n<p>\u8db3\u3057\u7b97\u3084\u5f15\u304d\u7b97\u306e\u5f0f\u3092\u5fae\u5206\u3059\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u548c\u3068\u5dee\u306e\u5fae\u5206\u306e\u516c\u5f0f<\/span><\/div><div class=\"cap_box_content\">\n<p>\\(y=f(x)+g(x)\\)\u3092\u5fae\u5206\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>\\[y^{\\prime}=f^{\\prime}(x)+g^{\\prime}(x)\\]<\/p>\n\n\n\n<p>\\(y=f(x)-g(x)\\)\u3092\u5fae\u5206\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>\\[y^{\\prime}=f^{\\prime}(x)-g^{\\prime}(x)\\]<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u4f8b\u984c<\/span><\/div><div class=\"cap_box_content\">\n<p>\\(y=2{x}^{3}+x^{2}\\)\u3092\u5fae\u5206\u3057\u306a\u3055\u3044\u3002<\/p>\n<\/div><\/div>\n\n\n\n<p>\u3053\u306e\u554f\u984c\u306f\\(2{x}^{3}\\)\u3068\\(x^{2}\\)\u3092\u305d\u308c\u305e\u308c\u8003\u3048\u307e\u3059\u3002<\/p>\n\n\n\n<p><span class=\"huto\">\\[y^\\prime=6{x}^2+2x\\]<\/span><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"koushiki3\">\u5b9a\u6570\u306e\u5fae\u5206<\/h3>\n\n\n\n<p>\\(x\\)\u3067\u5fae\u5206\u3059\u308b\u3068\u304d\u3001\\(x\\)\u3092\u6301\u305f\u306a\u3044\u5b9a\u6570\u306f\u5fae\u5206\u3059\u308b\u30680\u306b\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u5b9a\u6570\u306e\u5fae\u5206<\/span><\/div><div class=\"cap_box_content\">\n<p>\u5b9a\u6570\u95a2\u6570c\u3092\u5fae\u5206\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>\\[c^{\\prime}=0\\]<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u4f8b\u984c<\/span><\/div><div class=\"cap_box_content\">\n<p>\\(y=3{x}^{2}-4x+2\\)\u3092\u5fae\u5206\u3057\u306a\u3055\u3044\u3002<\/p>\n<\/div><\/div>\n\n\n\n<p>\\(3{x}^{2}-4x\\)\u306f\u3053\u308c\u307e\u3067\u540c\u69d8\u306b\u5fae\u5206\u3059\u308b\u3068\u3001\\(y=6x-4\\)\u306b\u306a\u308b\u3053\u3068\u306f\u5206\u304b\u308a\u307e\u3059\u306d\u3002\u6b8b\u308a\u306e2\u306f\\(x\\)\u3092\u6301\u305f\u306a\u3044\u5b9a\u6570\u306a\u306e\u30670\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p><span class=\"huto\">\\[y^{\\prime}=6x-4\\]<\/span><\/p>\n\n\n\n<p>\u5b9a\u6570c\u3092\u5fae\u5206\u3059\u308b\u3068\u3001\u306a\u305c0\u306b\u306a\u308b\u306e\u304b\u3092\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\\(c\\)\u3092\u7121\u7406\u3084\u308a\\(c x^{0}\\)\u3060\u3068\u8003\u3048\u308c\u3070\u3001\u5fae\u5206\u3057\u305f\u3068\u304d\u306b\\(c \\times 0\\)\u306a\u306e\u30670\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"koushiki4\">\u7a4d\u306e\u5fae\u5206<\/h3>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u7a4d\u306e\u5fae\u5206<\/span><\/div><div class=\"cap_box_content\">\n<p>\\[{f(x)g(x)}^{\\prime}=f^{\\prime}(x)g(x)+f(x)g^{\\prime}(x)\\]<\/p>\n<\/div><\/div>\n\n\n\n<p>\u7a4d\u306e\u5fae\u5206\u306e\u516c\u5f0f\u306f\u3053\u306e\u307e\u307e\u3060\u3068\u899a\u3048\u306b\u304f\u3044\u3068\u601d\u3044\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u79c1\u306f<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span style=\"font-size: 120%;\">\uff08\u305d\u306e\u307e\u307e\uff09\uff08\u5fae\u5206\uff09\uff0b\uff08\u5fae\u5206\uff09\uff08\u305d\u306e\u307e\u307e\uff09<\/span><\/p>\n\n\n\n<p>\u3068\u899a\u3048\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u4f8b\u984c<\/span><\/div><div class=\"cap_box_content\">\n<p>\\(y=(x^{2}-3x-1)(x^{2}+4)\\)\u3092\u5fae\u5206\u3057\u306a\u3055\u3044\u3002<\/p>\n<\/div><\/div>\n\n\n\n<p>\u89e3\u7b54<\/p>\n\n\n\n<p>\\begin{eqnarray}<br>y^{\\prime}&amp;=&amp;(x^{2}-3x-1)(x^{2}+4)\\\\<br>&amp;=&amp;(2x-3)(x^{2}+4)+(x^{2}-3x-1)\\bullet 2x\\\\<br>&amp;=&amp;2x^{3}+8x-3x^{2}-12+2x^{3}-6x^{2}-2x\\\\<br>&amp;=&amp;4x^{3}-9x^{2}+6x-12<br>\\end{eqnarray}<\/p>\n\n\n\n<p>\u7b54\u3048\u3000<span class=\"huto\">\\(y^{\\prime}=4x^{3}-9x^{2}+6x-12\\)<\/span><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"koushiki5\">\u5546\u306e\u5fae\u5206<\/h3>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u5546\u306e\u5fae\u5206<\/span><\/div><div class=\"cap_box_content\">\n<p>\\[\\displaystyle {\\frac{f(x)}{g(x)}}^{\\prime}=\\frac{f^{\\prime}(x)g(x)-f(x)g^{\\prime}(x)}{g(x)^{2}}\\]<\/p>\n<\/div><\/div>\n\n\n\n<p>\u7a4d\u306e\u5fae\u5206\u306e\u516c\u5f0f\u3067\u306f\u8db3\u3057\u7b97\u3067\u3057\u305f\u304c\u3001\u5546\u306e\u5fae\u5206\u306e\u516c\u5f0f\u306f\u5206\u5b50\u304c\u5f15\u304d\u7b97\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<p><span class=\"huto\">\u300c\u5206\u6bcd\u306f2\u4e57\u3001\u5206\u5b50\u306f\u7a4d\u306e\u5fae\u5206\u306e\u5f15\u304d\u7b97\u30d0\u30fc\u30b8\u30e7\u30f3\u300d<\/span>\u3068\u8003\u3048\u308b\u3068\u899a\u3048\u3084\u3059\u3044\u3067\u3059\u306d\u3002<\/p>\n\n\n\n<div class=\"swell-block-capbox cap_box\"><div class=\"cap_box_ttl\"><span>\u4f8b\u984c<\/span><\/div><div class=\"cap_box_content\">\n<p>\\(\\displaystyle y=\\frac{x^{2}-2}{2x+1}\\)\u3092\u5fae\u5206\u3057\u306a\u3055\u3044\u3002<\/p>\n<\/div><\/div>\n\n\n\n<p>\\begin{eqnarray}<br>\\displaystyle y^{\\prime}&amp;=&amp;\\frac{2x(2x+1)-(x^{2}-2x) \\bullet 2}{(2x+1)^{2}}\\\\<br>\\displaystyle &amp;=&amp; \\frac{2(x^{2}-x-1)}{(2x+1)^{2}}<br>\\end{eqnarray}<\/p>\n\n\n\n<p>\u7b54\u3048\u3000<span class=\"huto\">\\frac{2(x^{2}+x+2)}{(2x+1)^{2}}<\/span><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"matome\">\\(x^{n}\\)\u306e\u5fae\u5206\u307e\u3068\u3081<\/h2>\n\n\n\n<p>\u4eca\u56de\u306f\\(x^{n}\\)\u306e\u5fae\u5206\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p>\u3053\u308c\u306f\u5fae\u5206\u7a4d\u5206\u3092\u5b66\u7fd2\u3057\u3066\u3044\u4e0a\u3067\u7d76\u5bfe\u306b\u6b20\u304b\u305b\u306a\u3044\u57fa\u672c\u516c\u5f0f\u3067\u3059\u306e\u3067\u3001\u4eca\u56de\u3067\u30de\u30b9\u30bf\u30fc\u3057\u3066\u5b9a\u671f\u30c6\u30b9\u30c8\u306b\u6d3b\u304b\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n<div class=\"wp-block-group is-style-bg_stripe\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p><br>\u25ce\\(x^{n}\\)\u306e\u5fae\u5206<\/p>\n\n\n\n<p>\\[(x^{n})^{\\prime} =n x^{n-1}\\]<\/p>\n\n\n\n<p>\u25ce\u5b9a\u6570\u500d\u306e\u5fae\u5206<\/p>\n\n\n\n<p>\\[y^{\\prime}=k f^{\\prime}(x)\\]<\/p>\n\n\n\n<p>\u25ce\u548c\u3068\u5dee\u306e\u5fae\u5206<\/p>\n\n\n\n<p>\\(y=f(x)+g(x)\\)\u3092\u5fae\u5206\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>\\[y^{\\prime}=f^{\\prime}(x)+g^{\\prime}(x)\\]<\/p>\n\n\n\n<p>\\(y=f(x)-g(x)\\)\u3092\u5fae\u5206\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>\\[y^{\\prime}=f^{\\prime}(x)-g^{\\prime}(x)\\]<\/p>\n\n\n\n<p>\u25ce\u7a4d\u306e\u5fae\u5206<\/p>\n\n\n\n<p>\\[{f(x)g(x)}^{\\prime}=f^{\\prime}(x)g(x)+f(x)g^{\\prime}(x)\\]<\/p>\n\n\n\n<p>\u25ce\u5546\u306e\u5fae\u5206<\/p>\n\n\n\n<p>\\[\\displaystyle {\\frac{f(x)}{g(x)}}^{\\prime}=\\frac{f^{\\prime}(x)g(x)-f(x)g^{\\prime}(x)}{g(x)^{2}}\\]<\/p>\n<\/div><\/div>\n\n\n\n<p>\u3053\u306e\u8a08\u7b97\u306f\u30b9\u30e0\u30fc\u30ba\u306b\u3067\u304d\u308b\u3088\u3046\u306b\u3057\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n<p>\u305c\u3072\u672c\u30b5\u30a4\u30c8\u3092\u6d3b\u7528\u3057\u3066\u3001\u5fae\u5206\u3078\u306e\u7406\u89e3\u3092\u6df1\u3081\u3066\u304f\u3060\u3055\u3044\u306d\uff01<\/p>\n\n\n\n<p>\u305d\u308c\u3067\u306f\u6700\u5f8c\u307e\u3067\u3054\u89a7\u3044\u305f\u3060\u304d\u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p>\u307f\u3093\u306a\u306e\u52aa\u529b\u304c\u5831\u308f\u308c\u307e\u3059\u3088\u3046\u306b\uff01<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5165\u529b\u3057\u305f\u95a2\u6570\u3092\u5fae\u5206\u3057\u307e\u3059\u3002 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