{"id":1119,"date":"2026-01-18T12:10:04","date_gmt":"2026-01-18T02:10:04","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1119"},"modified":"2026-01-31T13:49:43","modified_gmt":"2026-01-31T03:49:43","slug":"year11-math-3-1-2-jp","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1119","title":{"rendered":"Year11-MATH-3-1-2 JP"},"content":{"rendered":"\n

Chapter 2: Introduction to Differential Calculus \u5fae\u5206\u7a4d\u5206\u5165\u9580<\/h4>\n\n\n\n

\u7b2c1\u7ae0\u3067\u306f\u3001\u9759\u7684\u306a\u95a2\u4fc2\u306b\u3064\u3044\u3066\u8003\u5bdf\u3057\u307e\u3057\u305f\u3002\u7b2c2\u7ae0\u3067\u306f\u3001\u5909\u5316\u306e\u6570\u5b66\u3067\u3042\u308b\u5fae\u7a4d\u5206\u5b66\u3078\u3068\u9032\u307f\u307e\u3059\u3002\u300c\u7269\u4f53\u306f\u3069\u3053\u306b\u3042\u308b\u306e\u304b\uff1f\u300d\u3068\u554f\u3046\u306e\u3067\u306f\u306a\u304f\u3001\u300c\u3053\u306e\u77ac\u9593\u3001\u7269\u4f53\u306f\u3069\u308c\u304f\u3089\u3044\u306e\u901f\u3055\u3067\u52d5\u3044\u3066\u3044\u308b\u306e\u304b\uff1f\u300d\u3068\u554f\u3046\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

2.1 The Concept of a Gradient \u52fe\u914d\u306e\u6982\u5ff5<\/h4>\n\n\n\n

\u7dda\u5f62\u95a2\u6570\uff08y<\/mi>=<\/mo>m<\/mi>x<\/mi>+<\/mo>c<\/mi><\/mrow>y = mx + c<\/annotation><\/semantics><\/math>\uff09\u3067\u306f\u3001\u52fe\u914d\u306f\u4e00\u5b9a\u3067\u3059\u3002\u76f4\u7dda\u4e0a\u306e\u3069\u3053\u306b\u3044\u3066\u3082\u3001\u300c\u6025\u5cfb\u3055\u300d\u306f\u540c\u3058\u3067\u3059\u3002\u3057\u304b\u3057\u3001f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>x<\/mi>2<\/mn><\/msup><\/mrow>f(x)=x^2<\/annotation><\/semantics><\/math>\u306e\u3088\u3046\u306a\u66f2\u7dda\u3067\u306f\u3001\u6025\u5cfb\u3055\u306f\u5e38\u306b\u5909\u5316\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

Secants vs. Tangents<\/h3>\n\n\n\n
    \n
  • Average Rate of Change (Secant):<\/strong>\u66f2\u7dda\u4e0a\u306e 2 <\/strong>\u3064\u306e\u7570\u306a\u308b\u70b9\u3092\u7d50\u3076\u7dda\u306e\u52fe\u914d\u3002<\/li>\n\n\n\n
  • Instantaneous Rate of Change (Tangent):<\/strong>\u66f2\u7dda\u306b\u4e00\u70b9\u3067\u63a5\u3059\u308b\u76f4\u7dda\u306e\u52fe\u914d\u3002\u3053\u308c\u304c\u5fae\u7a4d\u5206\u306e\u6838\u5fc3\u3067\u3059\u3002<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    2.2 The Derivative Function \u5fae\u5206\u95a2\u6570<\/h4>\n\n\n\n

    \u300c\u5fae\u5206\u300d\u3068\u306f\u3001\u4efb\u610f\u306ex\u5024\u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u50be\u304d\u3092\u8868\u3059\u5f0f\u3067\u3059\u3002\u4e3b\u306b2\u7a2e\u985e\u306e\u8868\u8a18\u6cd5\u3092\u7528\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

      \n
    1. Leibniz\u2019s Notation <\/strong>\u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u8a18\u6cd5:<\/strong> \u200bd<\/mi>y<\/mi><\/mrow>d<\/mi>x<\/mi><\/mrow><\/mfrac>\\frac{dy}{dx}<\/annotation><\/semantics><\/math> (\u300cy\u306ex\u306b\u95a2\u3059\u308b\u5fae\u5206\u300d\u3068\u8aad\u3080).<\/li>\n\n\n\n
    2. Lagrange\u2019s Notation <\/strong>\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u8868\u8a18\u6cd5:<\/strong> f<\/mi>\u2032<\/mtext>(<\/mo>x<\/mi>)<\/mo><\/mrow>f\u2032(x)<\/annotation><\/semantics><\/math> (read as “f prime of x”).<\/li>\n<\/ol>\n\n\n\n
      \n\n\n\n

      2.3 The Power Rule \u3079\u304d\u4e57\u5247<\/h4>\n\n\n\n

      \u3079\u304d\u4e57\u5247\u306f\u6570\u5b66\u7684\u624b\u6cd5\u306b\u304a\u3044\u3066\u6700\u3082\u91cd\u8981\u306a\u8fd1\u9053\u3067\u3059\u3002\u8907\u96d1\u306a\u300c\u7b2c\u4e00\u539f\u7406\u300d\u306e\u9650\u754c\u306b\u983c\u308b\u3053\u3068\u306a\u304f\u3001\u3042\u3089\u3086\u308b\u591a\u9805\u5f0f\u95a2\u6570\u306e\u5fae\u5206\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

      The Power Rule<\/strong><\/h3>\n\n\n\n
      \n

      If f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>x<\/mi>n<\/mi><\/msup><\/mrow>f(x)=x^{n}<\/annotation><\/semantics><\/math>, then: f<\/mi>\u2032<\/mtext>(<\/mo>x<\/mi>)<\/mo>=<\/mo>n<\/mi>x<\/mi>n<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup><\/mrow>f\u2032(x)=nx^{n\u22121}<\/annotation><\/semantics><\/math><\/p>\n<\/blockquote>\n\n\n\n

      Steps to differentiate:<\/strong><\/p>\n\n\n\n

        \n
      1. Multiply<\/strong> the coefficient by the current power (n). \u4fc2\u6570\u3092\u73fe\u5728\u306e\u3079\u304d\u4e57\uff08n<\/mi>n<\/annotation><\/semantics><\/math>\uff09\u306b\u639b\u3051\u307e\u3059\u3002<\/li>\n\n\n\n
      2. Subtract<\/strong> 1 from the power. \u3079\u304d\u4e57\u304b\u30891\u3092\u5f15\u304d\u307e\u3059\u3002<\/li>\n<\/ol>\n\n\n\n

        \u3079\u304d\u4e57\u5247\uff08\u3079\u304d\u3058\u3087\u3046\u305d\u304f\uff09\u3068\u306f\u3001\u3042\u308b\u91cf\u304c\u5225\u306e\u91cf\u306e\u3079\u304d\u4e57\u306b\u6bd4\u4f8b\u3059\u308b\uff08 <\/strong>Y<\/mi>=<\/mo>a<\/mi>X<\/mi>k<\/mi><\/msup><\/mrow>Y = aX^k<\/annotation><\/semantics><\/math>\u306e\u5f62\u3067\u8868\u3055\u308c\u308b\uff09\u3068\u3044\u3046\u95a2\u4fc2<\/strong>\u3067\u3001\u81ea\u7136\u754c\u3084\u793e\u4f1a\u73fe\u8c61\u3067\u5e83\u304f\u898b\u3089\u308c\u308b\u6cd5\u5247\u3067\u3059\u3002\u5c11\u6570\u306e\u8981\u7d20\u304c\u6975\u3081\u3066\u5927\u304d\u306a\u5024\uff08\u5f71\u97ff\u529b\u3084\u983b\u5ea6\u306a\u3069\uff09\u3092\u6301\u3061\u3001\u591a\u304f\u306e\u8981\u7d20\u306f\u5c0f\u3055\u3044\u5024\u306b\u7559\u307e\u308b\u3068\u3044\u3046\u3001\u300c\u4e00\u90e8\u306b\u504f\u308b\u300d<\/strong>\u975e\u5bfe\u79f0\u306a\u5206\u5e03\u3092\u793a\u3059\u306e\u304c\u7279\u5fb4\u3067\u3001SNS\u306e\u30d5\u30a9\u30ed\u30ef\u30fc\u6570\u3001\u90fd\u5e02\u306e\u4eba\u53e3\u3001\u5730\u9707\u306e\u5927\u304d\u3055\uff08\u30b0\u30fc\u30c6\u30f3\u30d9\u30eb\u30b0\u30fb\u30ea\u30d2\u30bf\u30fc\u5247\uff09\u3001\u53ce\u5165\u5206\u5e03\uff08\u30d1\u30ec\u30fc\u30c8\u306e\u6cd5\u5247\uff09\u306a\u3069\u3067\u89b3\u6e2c\u3055\u308c\u307e\u3059\u3002 <\/p>\n\n\n\n

        \u3079\u304d\u4e57\u5247\u306e\u4e3b\u306a\u7279\u5fb4<\/strong> <\/p>\n\n\n\n

          \n
        • \u6570\u5f0f\u8868\u73fe<\/strong>: Y<\/mi>=<\/mo>a<\/mi>X<\/mi>k<\/mi><\/msup><\/mrow>Y = aX^k<\/annotation><\/semantics><\/math>( a<\/mi>a<\/annotation><\/semantics><\/math> \u306f\u5b9a\u6570\u3001k<\/mi>k<\/annotation><\/semantics><\/math> \u306f\u3079\u304d\u6307\u6570\uff09\u3002<\/li>\n\n\n\n
        • \u5206\u5e03\u306e\u504f\u308a<\/strong>: \u5c11\u6570\u306e\u300c\u30d3\u30c3\u30b0\u300d\u306a\u4e8b\u8c61\u3068\u3001\u591a\u6570\u306e\u300c\u30b9\u30e2\u30fc\u30eb\u300d\u306a\u4e8b\u8c61\u304c\u5b58\u5728\u3059\u308b\u3002<\/li>\n\n\n\n
        • \u4e21\u5bfe\u6570\u30b0\u30e9\u30d5\u3067\u306e\u76f4\u7dda\u6027<\/strong>: \u30b0\u30e9\u30d5\u3092\u4e21\u5bfe\u6570\uff08\u5bfe\u6570-\u5bfe\u6570\uff09\u30b9\u30b1\u30fc\u30eb\u3067\u63cf\u304f\u3068\u3001\u76f4\u7dda\u95a2\u4fc2\u3068\u3057\u3066\u73fe\u308c\u308b\u3002<\/li>\n\n\n\n
        • \u666e\u904d\u6027<\/strong>: \u7570\u306a\u308b\u5206\u91ce\uff08\u7269\u7406\u3001\u7d4c\u6e08\u3001\u751f\u7269\u5b66\u306a\u3069\uff09\u306e\u73fe\u8c61\u3067\u3001\u540c\u3058\u3088\u3046\u306a\u30d1\u30bf\u30fc\u30f3\u304c\u898b\u3089\u308c\u308b\u3053\u3068\u304c\u3042\u308b\u3002 <\/li>\n<\/ul>\n\n\n\n

          \u8eab\u8fd1\u306a\u4f8b<\/strong> <\/p>\n\n\n\n

            \n
          • SNS\u306e\u30d5\u30a9\u30ed\u30ef\u30fc\u6570<\/strong>: \u4e00\u90e8\u306e\u6709\u540d\u4eba\u304c\u81a8\u5927\u306a\u30d5\u30a9\u30ed\u30ef\u30fc\u3092\u6301\u3061\u3001\u307b\u3068\u3093\u3069\u306e\u4eba\u306f\u5c11\u306a\u3044\u30d5\u30a9\u30ed\u30ef\u30fc\u6570\u3002<\/li>\n\n\n\n
          • \u90fd\u5e02\u306e\u4eba\u53e3<\/strong>: \u4e00\u90e8\u306e\u5927\u90fd\u5e02\u306b\u4eba\u53e3\u304c\u96c6\u4e2d\u3057\u3001\u591a\u304f\u306e\u5730\u65b9\u90fd\u5e02\u306f\u4eba\u53e3\u304c\u5c11\u306a\u3044\u3002<\/li>\n\n\n\n
          • \u5730\u9707\u306e\u898f\u6a21<\/strong>: \u5c0f\u3055\u306a\u5730\u9707\u306f\u983b\u7e41\u306b\u8d77\u3053\u308b\u304c\u3001\u5de8\u5927\u5730\u9707\u306f\u975e\u5e38\u306b\u307e\u308c\u3002<\/li>\n\n\n\n
          • \u5358\u8a9e\u306e\u51fa\u73fe\u983b\u5ea6<\/strong>: \u4f7f\u7528\u983b\u5ea6\u306e\u9ad8\u3044\u5358\u8a9e\u306f\u5c11\u306a\u304f\u3001\u4f4e\u3044\u5358\u8a9e\u306f\u975e\u5e38\u306b\u591a\u3044\uff08 Zipf\u306e\u6cd5\u5247\u3068\u3057\u3066\u77e5\u3089\u308c\u308b\uff09\u3002 <\/li>\n<\/ul>\n\n\n\n

            \u306a\u305c\u91cd\u8981\u304b<\/strong>
            \u3079\u304d\u4e57\u5247\u3092\u7406\u89e3\u3059\u308b\u3053\u3068\u3067\u3001\u30c7\u30fc\u30bf\u306b\u96a0\u3055\u308c\u305f\u69cb\u9020\u3084\u3001\u30b7\u30b9\u30c6\u30e0\u5168\u4f53\u306e\u632f\u308b\u821e\u3044\u3092\u4e88\u6e2c\u30fb\u5206\u6790\u3059\u308b\u306e\u306b\u5f79\u7acb\u3061\u307e\u3059\u3002\u8907\u96d1\u306a\u30b7\u30b9\u30c6\u30e0\u304c\u968e\u5c64\u7684<\/strong><\/strong>\u3067\u3042\u308a\u3001\u7279\u5b9a\u306e\u666e\u904d\u7684\u306a\u4ed5\u7d44\u307f<\/strong>\u306b\u3088\u3063\u3066\u652f\u3048\u3089\u308c\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u5506\u3059\u308b\u91cd\u8981\u306a\u6982\u5ff5\u3067\u3059\u3002\u00a0<\/p>\n\n\n\n

            ****************<\/p>\n\n\n\n

            \u30aa\u30fc\u30b9\u30c8\u30e9\u30ea\u30a2\u306e11\u5e74\u751f\u6570\u5b66\u6307\u5c0e\u6cd5\uff08\u304a\u3088\u3073\u5c02\u9580\u79d1\u76ee\uff09\u3067\u306f\u3001\u3079\u304d\u4e57\u5247\u306f\u3079\u304d\u95a2\u6570\uff08x<\/mi>n<\/mi><\/msup>x^n<\/annotation><\/semantics><\/math>\uff09\u306e\u5fae\u5206\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u57fa\u672c\u7684\u304b\u3064\u6700\u3082\u4e00\u822c\u7684\u306b\u7528\u3044\u3089\u308c\u308b\u8fd1\u9053\u3067\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u751f\u5f92\u306f\u9762\u5012\u306a\u300c\u7b2c\u4e00\u539f\u7406\u300d\u306b\u3088\u308b\u6975\u9650\u5b9a\u7fa9\u3092\u6bce\u56de\u4f7f\u7528\u3059\u308b\u3053\u3068\u306a\u304f\u3001\u52fe\u914d\u95a2\u6570 d<\/mi>y<\/mi><\/mrow>d<\/mi>x<\/mi><\/mrow><\/mfrac>\\frac{dy}{dx}<\/annotation><\/semantics><\/math> (\u307e\u305f\u306f\uff09f<\/mi>\u2018<\/mtext>(<\/mo>x<\/mi>)<\/mo><\/mrow>f\u2018(x)<\/annotation><\/semantics><\/math> \u3092\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

            >> The Power Rule Formula <\/strong><\/p>\n\n\n\n

            If f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>x<\/mi>n<\/mi><\/msup><\/mrow>f(x)=x^n<\/annotation><\/semantics><\/math> , then the derivative is: d<\/mi>d<\/mi>x<\/mi><\/mrow><\/mfrac>(<\/mo>x<\/mi>n<\/mi><\/msup>)<\/mo>=<\/mo>n<\/mi>x<\/mi>n<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup><\/mrow> \\frac{d}{dx}(x^n)=nx^{n\u22121}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

            Where <\/em>is any real number.<\/em><\/p>\n\n\n\n

            If the term has a constant coefficient , such as f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>a<\/mi>x<\/mi>n<\/mi><\/msup><\/mrow>f(x)=ax^n<\/annotation><\/semantics><\/math> , the rule is: d<\/mi>d<\/mi>x<\/mi><\/mrow><\/mfrac>(<\/mo>a<\/mi>x<\/mi>n<\/mi><\/msup>)<\/mo><\/mrow>\\frac{d}{dx}(ax^n)<\/annotation><\/semantics><\/math> = a<\/mi>n<\/mi>x<\/mi>n<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup><\/mrow>anx^{n-1}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

            Multiply the exponent by the coefficient, then subtract 1 from the exponent.<\/em> <\/p>\n\n\n\n

            >> Step-by-Step Application<\/strong> <\/p>\n\n\n\n

              \n
            1. Identify <\/strong>n<\/mi>n<\/annotation><\/semantics><\/math>: <\/strong>Look at the power of x<\/mi>x<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
            2. Bring it down:<\/strong> Multiply the entire term by this power (n<\/mi>n<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n
            3. Reduce it:<\/strong> Subtract 1 from the original power (n<\/mi>\u2212<\/mo>1<\/mn><\/mrow>n-1<\/annotation><\/semantics><\/math>). <\/li>\n<\/ol>\n\n\n\n

              Example 1: Simple Power<\/strong><\/p>\n\n\n\n

              y = <\/strong>X<\/mi>5<\/mn><\/msup>X^5<\/annotation><\/semantics><\/math> <\/p>\n\n\n\n

              d<\/mi>y<\/mi><\/mrow>d<\/mi>x<\/mi><\/mrow><\/mfrac>=<\/mo>5<\/mn>x<\/mi>5<\/mn>\u2013<\/mtext>1<\/mn><\/mrow><\/msup>=<\/mo>5<\/mn>x<\/mi>4<\/mn><\/msup><\/mrow>\\frac{dy}{dx} = 5x^{5\u20131} = 5x^4<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

              Example 2: Coefficient and Power<\/strong> <\/p>\n\n\n\n

              y<\/mi>=<\/mo>4<\/mn>x<\/mi>3<\/mn><\/msup><\/mrow>y = 4x^3<\/annotation><\/semantics><\/math> <\/p>\n\n\n\n

              d<\/mi>y<\/mi><\/mrow>d<\/mi>x<\/mi><\/mrow><\/mfrac>=<\/mo>(<\/mo>3<\/mn>x<\/mi>4<\/mn>)<\/mo>x<\/mi>3<\/mn>\u2212<\/mo>1<\/mn><\/mrow><\/msup>=<\/mo>12<\/mn>x<\/mi>2<\/mn><\/msup><\/mrow>\\frac {dy}{dx} = (3×4)x^{3-1} = 12x^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

              >> <\/strong>Key Applications in Year 11<\/strong><\/p>\n\n\n\n

              \u591a\u9805\u5f0f: \u3079\u304d\u4e57\u5247\u306f\u5404\u9805\u306b\u500b\u5225\u306b\u9069\u7528\u3055\u308c\u307e\u3059 (\u548c\u5247)\u3002<\/strong><\/p>\n\n\n\n

                \n
              • Example: <\/em>d<\/mi>d<\/mi>x<\/mi><\/mrow><\/mfrac>(<\/mo>3<\/mn>x<\/mi>2<\/mn><\/msup>+<\/mo>5<\/mn>x<\/mi>+<\/mo>2<\/mn>)<\/mo>=<\/mo>6<\/mn>x<\/mi>+<\/mo>5<\/mn><\/mrow>\\frac{d}{dx}(3x^{2}+5x+2)=6x+5 <\/annotation><\/semantics><\/math>. <\/li>\n\n\n\n
              • \u8ca0\u306e\u6307\u6570\uff08\u9006\u6570\uff09<\/strong>:<\/strong> \u5206\u6bcd\u306e\u9805\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/li>\n\n\n\n
              • Example: <\/em>d<\/mi>d<\/mi>x<\/mi><\/mrow><\/mfrac>(<\/mo>1<\/mn>x<\/mi>2<\/mn><\/msup><\/mfrac>)<\/mo>=<\/mo>d<\/mi>d<\/mi>x<\/mi><\/mrow><\/mfrac>(<\/mo>x<\/mi>\u2212<\/mo>2<\/mn><\/mrow><\/msup>)<\/mo>=<\/mo>\u2212<\/mo>2<\/mn>x<\/mi>\u2212<\/mo>3<\/mn><\/mrow><\/msup>=<\/mo>\u2212<\/mo>2<\/mn>x<\/mi>3<\/mn><\/msup><\/mfrac><\/mrow>\\frac{d}{dx}(\\frac{1}{x^{2}})=\\frac{d}{dx}(x^{-2})=-2x^{-3}=-\\frac{2}{x^{3}} <\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
              • \u5206\u6570\u6307\u6570\uff08\u6839\uff09<\/strong>\uff1a\u5e73\u65b9\u6839\u306a\u3069\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/li>\n\n\n\n
              • Example:<\/em> d<\/mi>d<\/mi>x<\/mi><\/mrow><\/mfrac>(<\/mo>x<\/mi><\/msqrt>)<\/mo>=<\/mo>d<\/mi>d<\/mi>x<\/mi><\/mrow><\/mfrac>(<\/mo>x<\/mi>1<\/mn>\/<\/mi>2<\/mn><\/mrow><\/msup>)<\/mo>=<\/mo>1<\/mn>2<\/mn><\/mfrac>x<\/mi>\u2212<\/mo>1<\/mn>\/<\/mi>2<\/mn><\/mrow><\/msup>=<\/mo>1<\/mn>2<\/mn>x<\/mi><\/msqrt><\/mrow><\/mfrac><\/mrow>\\frac{d}{dx}(\\sqrt{x})=\\frac{d}{dx}(x^{1\/2})=\\frac{1}{2}x^{-1\/2}=\\frac{1}{2\\sqrt{x}}<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
              • \u5b9a\u6570\u30eb\u30fc\u30eb: \u5b9a\u6570\u306e\u5c0e\u95a2\u6570\u306f 0 \u3067\u3059\u3002<\/li>\n\n\n\n
              • Example: <\/em>d<\/mi>d<\/mi>x<\/mi><\/mrow><\/mfrac>(<\/mo>10<\/mn>)<\/mo>=<\/mo>0<\/mn><\/mrow>\\frac{d}{dx}(10)=0<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
              • \u7dda\u5f62\u9805: \u306e\u5c0e\u95a2\u6570\u306f a<\/mi>a<\/annotation><\/semantics><\/math>\u3067\u3059\u3002<\/li>\n\n\n\n
              • Example: <\/em>d<\/mi>d<\/mi>x<\/mi><\/mrow><\/mfrac>(<\/mo>7<\/mn>x<\/mi>)<\/mo>=<\/mo>7<\/mn><\/mrow>\\frac{d}{dx}(7x)=7<\/annotation><\/semantics><\/math>.<\/em><\/li>\n<\/ul>\n\n\n\n

                >> Key Requirements & Common Pitfalls \u4e3b\u306a\u8981\u4ef6\u3068\u3088\u304f\u3042\u308b\u843d\u3068\u3057\u7a74<\/p>\n\n\n\n

                  \n
                • \u6e96\u5099: \u3079\u304d\u4e57\u5247\u3092\u9069\u7528\u3059\u308b\u524d\u306b\u3001\u6307\u6570\u898f\u5247 ( <\/strong>1<\/mn>\/<\/mi>x<\/mi>n<\/mi><\/msup>=<\/mo>x<\/mi>\u2212<\/mo>n<\/mi><\/mrow><\/msup><\/mrow>1\/x^{n}=x^{-n}<\/annotation><\/semantics><\/math> \u304a\u3088\u3073 <\/strong>x<\/mi>n<\/mi><\/msup>m<\/mi><\/mroot>=<\/mo>x<\/mi>n<\/mi>\/<\/mi>m<\/mi><\/mrow><\/msup><\/mrow>\\sqrt[m]{x^{n}}=x^{n\/m}<\/annotation><\/semantics><\/math>) \u3092\u4f7f\u7528\u3057\u3066\u3001\u5206\u6bcd\u5185\u307e\u305f\u306f\u30eb\u30fc\u30c8\u8a18\u53f7\u306e\u4e0b\u306e\u5f0f\u3092\u66f8\u304d\u76f4\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/strong><\/li>\n\n\n\n
                • \u8ca0\u306e\u6570: \u8ca0\u306e\u6307\u6570\u306e\u5834\u5408\u306f\u3001\u8ca0\u306e\u5024\u304c\u5927\u304d\u304f\u306a\u308b\u3053\u3068\u3092\u899a\u3048\u3066\u304a\u3044\u3066\u304f\u3060\u3055\u3044 (\u305f\u3068\u3048\u3070 \u3001<\/strong>x<\/mi>\u2212<\/mo>2<\/mn><\/mrow><\/msup>x^{-2}<\/annotation><\/semantics><\/math> \u306e\u5c0e\u95a2\u6570\u306f <\/strong>\u2212<\/mo>2<\/mn>x<\/mi>\u2212<\/mo>3<\/mn><\/mrow><\/msup><\/mrow>-2x^{-3}<\/annotation><\/semantics><\/math> \u3067\u3059)\u3002<\/strong> <\/li>\n\n\n\n
                • \u7c21\u7565\u5316<\/strong>: \u4e57\u7b97\u5f8c\u306f\u5fc5\u305a\u6570\u5024\u4fc2\u6570\u3092\u7c21\u7565\u5316\u3057\u307e\u3059\u3002<\/li>\n<\/ul>\n\n\n\n

                  \u6ce8: NSW HSC \u304a\u3088\u3073\u30aa\u30fc\u30b9\u30c8\u30e9\u30ea\u30a2\u306e\u4ed6\u306e\u7ba1\u8f44\u533a\u57df\u3067\u306f\u3001\u3079\u304d\u4e57\u5247\u306f\u901a\u5e38\u3001\u9650\u754c\u3068\u7b2c\u4e00\u539f\u7406\u306e\u7814\u7a76\u306b\u7d9a\u3044\u3066\u3001\u6b63\u5f0f\u306a\u5dee\u5225\u5316\u3078\u306e\u7b2c\u4e00\u6b69\u3068\u3057\u3066 11 \u5e74\u751f\u3067\u5c0e\u5165\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

                    \n
                  • *******************************************<\/li>\n<\/ul>\n\n\n\n

                    Constants and Linear Terms \u5b9a\u6570\u3068\u7dda\u5f62\u9805<\/h4>\n\n\n\n
                      \n
                    • Constants:<\/strong> The derivative of a constant \u5b9a\u6570\uff1a\u5b9a\u6570\u306e\u5c0e\u95a2\u6570 (e.g., f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>5<\/mn><\/mrow>f(x)=5<\/annotation><\/semantics><\/math>) is 0<\/strong>. (A flat line has no steepness).<\/li>\n\n\n\n
                    • Linear Terms:<\/strong> The derivative of f(x)=ax is simply a<\/strong>. \u7dda\u5f62\u9805\uff1af(x)=ax\u306e\u5c0e\u95a2\u6570\u306f\u5358\u7d14\u306ba<\/li>\n<\/ul>\n\n\n\n
                      \n

                      Worked Example 1: Differentiating a Polynomial<\/strong> Differentiate.f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>4<\/mn>x<\/mi>3<\/mn><\/msup>\u2212<\/mo>2<\/mn>x<\/mi>2<\/mn><\/msup>+<\/mo>5<\/mn>x<\/mi>\u2212<\/mo>7<\/mn><\/mrow>f(x) = 4x^3 – 2x^2 + 5x – 7<\/annotation><\/semantics><\/math><\/p>\n<\/blockquote>\n\n\n\n
                        \n
                      1. Term 1 (4<\/mn>x<\/mi>3<\/mn><\/msup><\/mrow>4x^3<\/annotation><\/semantics><\/math>): 3\u00d74=12, then 3\u22121=2. Result: .<\/li>\n\n\n\n
                      2. Term 2 (\u2212<\/mo>2<\/mn>x<\/mi>2<\/mn><\/msup><\/mrow>\u22122x^2<\/annotation><\/semantics><\/math>): 2\u00d7\u22122=\u22124, then 2\u22121=1. Result: \u22124x.<\/li>\n\n\n\n
                      3. Term 3 (5<\/mn>x<\/mi><\/mrow>5x<\/annotation><\/semantics><\/math>): Gradient of is simply 5.<\/li>\n\n\n\n
                      4. Term 4 (\u22127): Derivative is 0.<\/li>\n<\/ol>\n\n\n\n
                        \n

                        Final Answer:<\/strong> f<\/mi>\u2032<\/mo><\/msup>(<\/mo>x<\/mi>)<\/mo>=<\/mo>12<\/mn>x<\/mi>2<\/mn><\/msup>\u2212<\/mo>4<\/mn>x<\/mi>+<\/mo>5<\/mn><\/mrow>f'(x) = 12x^2 – 4x + 5<\/annotation><\/semantics><\/math><\/p>\n<\/blockquote>\n\n\n\n
                        \n\n\n\n

                        2.4 Finding the Gradient at a Point \u70b9\u306b\u304a\u3051\u308b\u52fe\u914d\u306e\u8a08\u7b97<\/h4>\n\n\n\n

                        Once you have the derivative function (f\u2032(x)), you can find the exact steepness of the curve at any x-coordinate by substituting the value into the derivative. \u5fae\u5206\u95a2\u6570 (f\u2032(x)) \u3092\u53d6\u5f97\u3057\u305f\u3089\u3001\u305d\u306e\u5024\u3092\u5fae\u5206\u306b\u4ee3\u5165\u3059\u308b\u3053\u3068\u3067\u3001\u4efb\u610f\u306e x \u5ea7\u6a19\u306b\u304a\u3051\u308b\u66f2\u7dda\u306e\u6b63\u78ba\u306a\u50be\u304d\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

                        \n

                        Worked Example 2: Finding a Specific Gradient<\/strong> Find the gradient of the curve at the point where .<\/p>\n<\/blockquote>\n\n\n\n

                          \n
                        1. Find the derivative: d<\/mi>y<\/mi><\/mrow>d<\/mi>x<\/mi><\/mrow><\/mfrac>\\frac{dy}{dx}<\/annotation><\/semantics><\/math> \u200b= 2x+3.<\/li>\n\n\n\n
                        2. Substitute x=2: 2(2)+3=7.<\/li>\n\n\n\n
                        3. Interpretation:<\/strong> At the point (2,10), the curve is rising at a rate of 7 units up for every 1 unit across.\u70b9\uff082,10\uff09\u3067\u306f\u3001\u66f2\u7dda\u306f1\u5358\u4f4d\u6a2a\u65b9\u5411\u306b\u4f38\u3073\u308b\u3054\u3068\u306b7\u5358\u4f4d\u4e0a\u6607\u3059\u308b\u3002<\/li>\n<\/ol>\n\n\n\n

                          2.5 Practice Problems \u7df4\u7fd2\u554f\u984c<\/h4>\n\n\n\n

                          Part A: Basic Differentiation \u57fa\u672c\u7684\u306a\u5fae\u5206<\/h4>\n\n\n\n

                          Differentiate the following with respect to x: \u6b21\u306e\u5f0f\u3092x\u306b\u3064\u3044\u3066\u5fae\u5206\u305b\u3088<\/p>\n\n\n\n

                          1. f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>x<\/mi>5<\/mn><\/msup><\/mrow>f(x) = x^5<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                          2.y<\/mi>=<\/mo>10<\/mn>x<\/mi>2<\/mn><\/msup>\u2212<\/mo>4<\/mn>x<\/mi><\/mrow>y = 10x^2 – 4x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                          3.g<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>1<\/mn>3<\/mn><\/mfrac>x<\/mi>3<\/mn><\/msup>+<\/mo>5<\/mn><\/mrow>g(x) = \\frac{1}{3}x^3 + 5<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                          4.f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>8<\/mn>x<\/mi>\u2212<\/mo>2<\/mn><\/mrow>f(x) = 8x – 2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                          Part B: Gradients at Points<\/h3>\n\n\n\n
                            \n
                          1. Find the gradient of f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>2<\/mn>x<\/mi>2<\/mn><\/msup>\u2212<\/mo>5<\/mn><\/mrow>f(x) = 2x^2 – 5<\/annotation><\/semantics><\/math> at x<\/mi>=<\/mo>3<\/mn><\/mrow>x = 3<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
                          2. For the curve y<\/mi>=<\/mo>x<\/mi>3<\/mn><\/msup>\u2212<\/mo>2<\/mn>x<\/mi><\/mrow>y = x^3 – 2x<\/annotation><\/semantics><\/math> , find the coordinates where the gradient is equal to 1.<\/li>\n\n\n\n
                          3. Challenge:<\/strong> If f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>a<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>4<\/mn>x<\/mi><\/mrow>f(x)=ax^2+4x<\/annotation><\/semantics><\/math> and f<\/mi>\u2032<\/mo><\/msup>(<\/mo>1<\/mn>)<\/mo>=<\/mo>10<\/mn><\/mrow>f'(1) = 10<\/annotation><\/semantics><\/math> , find the value of a<\/mi>a<\/annotation><\/semantics><\/math>.<\/li>\n<\/ol>\n\n\n\n
                            \n\n\n\n

                            Solutions (Summary)<\/h3>\n\n\n\n
                              \n
                            • 1. 5<\/mn>x<\/mi>4<\/mn><\/msup><\/mrow>5x^4<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
                            • 2.<\/strong> 20<\/mn>x<\/mi>\u2212<\/mo>4<\/mn><\/mrow>20x – 4<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
                            • 3.<\/strong> x<\/mi>2<\/mn><\/msup>x^2<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
                            • 4.<\/strong> 8<\/li>\n\n\n\n
                            • 5.<\/strong> f<\/mi>\u2032<\/mo><\/msup>(<\/mo>x<\/mi>)<\/mo>=<\/mo>4<\/mn>x<\/mi><\/mrow>f'(x) = 4x<\/annotation><\/semantics><\/math>. At x<\/mi>=<\/mo>3<\/mn><\/mrow>x=3<\/annotation><\/semantics><\/math>, gradient = 12.<\/li>\n\n\n\n
                            • 6.<\/strong> d<\/mi>y<\/mi><\/mrow>d<\/mi>x<\/mi><\/mrow><\/mfrac>=<\/mo>3<\/mn>x<\/mi>2<\/mn><\/msup>\u2212<\/mo>2<\/mn><\/mrow>\\frac{dy}{dx} = 3x^2 – 2<\/annotation><\/semantics><\/math>. Set to 1: 3<\/mn>x<\/mi>2<\/mn><\/msup>\u2212<\/mo>2<\/mn>=<\/mo>1<\/mn>\u2192<\/mo>3<\/mn>x<\/mi>2<\/mn><\/msup>=<\/mo>3<\/mn>\u2192<\/mo>x<\/mi>=<\/mo>\u00b1<\/mo>1<\/mn><\/mrow>3x^2 – 2 = 1 \\rightarrow 3x^2 = 3 \\rightarrow x = \\pm 1<\/annotation><\/semantics><\/math>. Points are (1,\u22121) and (\u22121,1).<\/li>\n\n\n\n
                            • 7.<\/strong> f<\/mi>\u2032<\/mo><\/msup>(<\/mo>x<\/mi>)<\/mo>=<\/mo>2<\/mn>a<\/mi>x<\/mi>+<\/mo>4<\/mn><\/mrow>f'(x) = 2ax + 4<\/annotation><\/semantics><\/math>. So, 2<\/mn>a<\/mi>(<\/mo>1<\/mn>)<\/mo>+<\/mo>4<\/mn>=<\/mo>10<\/mn>\u2192<\/mo>2<\/mn>a<\/mi>=<\/mo>6<\/mn>\u2192<\/mo>a<\/mi>=<\/mo>3<\/mn><\/mrow>2a(1)+4=10\u21922a=6\u2192a=3<\/annotation><\/semantics><\/math>.<\/li>\n<\/ul>\n\n\n\n
                              \n\n\n\n



                              <\/h3>\n","protected":false},"excerpt":{"rendered":"

                              Chapter 2: Introduction to Differential Calculus \u5fae\u5206\u7a4d\u5206\u5165\u9580 \u7b2c1\u7ae0\u3067\u306f\u3001\u9759\u7684\u306a\u95a2\u4fc2\u306b\u3064\u3044\u3066\u8003\u5bdf\u3057\u307e\u3057\u305f\u3002\u7b2c2\u7ae0\u3067\u306f\u3001\u5909\u5316\u306e\u6570\u5b66\u3067\u3042\u308b\u5fae\u7a4d\u5206\u5b66\u3078\u3068\u9032\u307f\u307e\u3059\u3002\u300c\u7269\u4f53\u306f\u3069\u3053\u306b\u3042\u308b\u306e\u304b\uff1f\u300d\u3068\u554f\u3046\u306e\u3067\u306f\u306a\u304f\u3001\u300c\u3053\u306e\u77ac\u9593\u3001\u7269\u4f53\u306f\u3069\u308c\u304f\u3089\u3044\u306e\u901f\u3055\u3067\u52d5\u3044\u3066\u3044\u308b\u306e\u304b\uff1f\u300d\u3068\u554f\u3046\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 2.1 The Concept of a Gradient \u52fe\u914d\u306e\u6982\u5ff5 \u7dda\u5f62\u95a2\u6570\uff08y=mx+cy = mx + c\uff09\u3067\u306f\u3001\u52fe\u914d\u306f\u4e00\u5b9a\u3067\u3059\u3002\u76f4\u7dda\u4e0a\u306e\u3069\u3053\u306b\u3044\u3066\u3082\u3001\u300c\u6025\u5cfb\u3055\u300d\u306f\u540c\u3058\u3067\u3059\u3002\u3057\u304b\u3057\u3001f(x)=x2f(x)=x^2\u306e\u3088\u3046\u306a\u66f2\u7dda\u3067\u306f\u3001\u6025\u5cfb\u3055\u306f\u5e38\u306b\u5909\u5316\u3057\u307e\u3059\u3002 Secants vs. Tangents 2.2 The Derivative Function \u5fae\u5206\u95a2\u6570 \u300c\u5fae\u5206\u300d\u3068\u306f\u3001\u4efb\u610f\u306ex\u5024\u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u50be\u304d\u3092\u8868\u3059\u5f0f\u3067\u3059\u3002\u4e3b\u306b2\u7a2e\u985e\u306e\u8868\u8a18\u6cd5\u3092\u7528\u3044\u307e\u3059\u3002 2.3 The Power Rule \u3079\u304d\u4e57\u5247 \u3079\u304d\u4e57\u5247\u306f\u6570\u5b66\u7684\u624b\u6cd5\u306b\u304a\u3044\u3066\u6700\u3082\u91cd\u8981\u306a\u8fd1\u9053\u3067\u3059\u3002\u8907\u96d1\u306a\u300c\u7b2c\u4e00\u539f\u7406\u300d\u306e\u9650\u754c\u306b\u983c\u308b\u3053\u3068\u306a\u304f\u3001\u3042\u3089\u3086\u308b\u591a\u9805\u5f0f\u95a2\u6570\u306e\u5fae\u5206\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 The Power Rule If f(x)=xnf(x)=x^{n}, then: f\u2032(x)=nxn\u22121f\u2032(x)=nx^{n\u22121} Steps to differentiate: \u3079\u304d\u4e57\u5247\uff08\u3079\u304d\u3058\u3087\u3046\u305d\u304f\uff09\u3068\u306f\u3001\u3042\u308b\u91cf\u304c\u5225\u306e\u91cf\u306e\u3079\u304d\u4e57\u306b\u6bd4\u4f8b\u3059\u308b\uff08 Y=aXkY = aX^k\u306e\u5f62\u3067\u8868\u3055\u308c\u308b\uff09\u3068\u3044\u3046\u95a2\u4fc2\u3067\u3001\u81ea\u7136\u754c\u3084\u793e\u4f1a\u73fe\u8c61\u3067\u5e83\u304f\u898b\u3089\u308c\u308b\u6cd5\u5247\u3067\u3059\u3002\u5c11\u6570\u306e\u8981\u7d20\u304c\u6975\u3081\u3066\u5927\u304d\u306a\u5024\uff08\u5f71\u97ff\u529b\u3084\u983b\u5ea6\u306a\u3069\uff09\u3092\u6301\u3061\u3001\u591a\u304f\u306e\u8981\u7d20\u306f\u5c0f\u3055\u3044\u5024\u306b\u7559\u307e\u308b\u3068\u3044\u3046\u3001\u300c\u4e00\u90e8\u306b\u504f\u308b\u300d\u975e\u5bfe\u79f0\u306a\u5206\u5e03\u3092\u793a\u3059\u306e\u304c\u7279\u5fb4\u3067\u3001SNS\u306e\u30d5\u30a9\u30ed\u30ef\u30fc\u6570\u3001\u90fd\u5e02\u306e\u4eba\u53e3\u3001\u5730\u9707\u306e\u5927\u304d\u3055\uff08\u30b0\u30fc\u30c6\u30f3\u30d9\u30eb\u30b0\u30fb\u30ea\u30d2\u30bf\u30fc\u5247\uff09\u3001\u53ce\u5165\u5206\u5e03\uff08\u30d1\u30ec\u30fc\u30c8\u306e\u6cd5\u5247\uff09\u306a\u3069\u3067\u89b3\u6e2c\u3055\u308c\u307e\u3059\u3002  \u3079\u304d\u4e57\u5247\u306e\u4e3b\u306a\u7279\u5fb4  \u8eab\u8fd1\u306a\u4f8b  \u306a\u305c\u91cd\u8981\u304b\u3079\u304d\u4e57\u5247\u3092\u7406\u89e3\u3059\u308b\u3053\u3068\u3067\u3001\u30c7\u30fc\u30bf\u306b\u96a0\u3055\u308c\u305f\u69cb\u9020\u3084\u3001\u30b7\u30b9\u30c6\u30e0\u5168\u4f53\u306e\u632f\u308b\u821e\u3044\u3092\u4e88\u6e2c\u30fb\u5206\u6790\u3059\u308b\u306e\u306b\u5f79\u7acb\u3061\u307e\u3059\u3002\u8907\u96d1\u306a\u30b7\u30b9\u30c6\u30e0\u304c\u968e\u5c64\u7684\u3067\u3042\u308a\u3001\u7279\u5b9a\u306e\u666e\u904d\u7684\u306a\u4ed5\u7d44\u307f\u306b\u3088\u3063\u3066\u652f\u3048\u3089\u308c\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u5506\u3059\u308b\u91cd\u8981\u306a\u6982\u5ff5\u3067\u3059\u3002\u00a0 **************** \u30aa\u30fc\u30b9\u30c8\u30e9\u30ea\u30a2\u306e11\u5e74\u751f\u6570\u5b66\u6307\u5c0e\u6cd5\uff08\u304a\u3088\u3073\u5c02\u9580\u79d1\u76ee\uff09\u3067\u306f\u3001\u3079\u304d\u4e57\u5247\u306f\u3079\u304d\u95a2\u6570\uff08xnx^n\uff09\u306e\u5fae\u5206\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u57fa\u672c\u7684\u304b\u3064\u6700\u3082\u4e00\u822c\u7684\u306b\u7528\u3044\u3089\u308c\u308b\u8fd1\u9053\u3067\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u751f\u5f92\u306f\u9762\u5012\u306a\u300c\u7b2c\u4e00\u539f\u7406\u300d\u306b\u3088\u308b\u6975\u9650\u5b9a\u7fa9\u3092\u6bce\u56de\u4f7f\u7528\u3059\u308b\u3053\u3068\u306a\u304f\u3001\u52fe\u914d\u95a2\u6570 dydx\\frac{dy}{dx} 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