{"id":1340,"date":"2026-01-27T15:00:21","date_gmt":"2026-01-27T05:00:21","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1340"},"modified":"2026-01-27T15:00:21","modified_gmt":"2026-01-27T05:00:21","slug":"year12-math-3-3-2-further-differentiation","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1340","title":{"rendered":"Year12 MATH 3-3-2 further differentiation"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In QLD Year 12 Mathematical Methods, &#8220;Further Differentiation and Applications&#8221; (specifically Topic 2 of Unit 3) focuses on <mark>expanding calculus techniques beyond basic polynomial differentiation to include transcendental functions (exponential, logarithmic, trigonometric) and advanced rules, alongside their practical applications<\/mark>.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here is a detailed breakdown of the key components of this topic:&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1. Differentiation Techniques (Formulas and Rules)&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This section involves mastering new derivative formulas and rules required to differentiate complex, composite functions:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Derivatives of Exponential Functions:<\/strong> Rules for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><msup><mi>e<\/mi><mi>x<\/mi><\/msup><annotation encoding=\"text\/plain\">e to the x-th power<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><msup><mi>e<\/mi><mrow><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><\/msup><annotation encoding=\"text\/plain\">e raised to the f of x power<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Derivatives of Logarithmic Functions:<\/strong> Rules for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>ln<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">l n x<\/annotation><\/semantics><\/math>and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>ln<\/mi><mo>(<\/mo><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">l n f of x<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Derivatives of Trigonometric Functions:<\/strong> Rules for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>sin<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">sine x<\/annotation><\/semantics><\/math>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>cos<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">cosine x<\/annotation><\/semantics><\/math>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>tan<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">tangent x<\/annotation><\/semantics><\/math> and their chain rule combinations.<\/li>\n\n\n\n<li><strong>Chain Rule:<\/strong> Used for composite functions, such as <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mo>[<\/mo><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><msup><mo>]<\/mo><mi>n<\/mi><\/msup><\/mrow><annotation encoding=\"text\/plain\">y equals open bracket f of x close bracket to the n-th power<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Product Rule:<\/strong> Used for differentiating the product of two functions (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>u<\/mi><mi>v<\/mi><\/mrow><annotation encoding=\"text\/plain\">u v<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Quotient Rule:<\/strong> Used for differentiating the quotient of two functions (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mfrac><mi>u<\/mi><mi>v<\/mi><\/mfrac><annotation encoding=\"text\/plain\">u over v end-fraction<\/annotation><\/semantics><\/math>).\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">2. Applications of Differentiation&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This involves using the above techniques to analyze functions and solve real-world problems:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Tangents and Normals:<\/strong> Finding the equations of tangent and normal lines to curves at specific points.<\/li>\n\n\n\n<li><strong>Stationary Points:<\/strong> Identifying and classifying maximum, minimum, and stationary points of inflection using first and second derivatives.<\/li>\n\n\n\n<li><strong>Curve Sketching:<\/strong> Using derivatives to determine increasing\/decreasing intervals, concavity (concave up\/down), and finding points of inflection.<\/li>\n\n\n\n<li><strong>Optimization:<\/strong> Solving practical problems to find maximum or minimum values (e.g., maximizing profit, minimizing surface area).<\/li>\n\n\n\n<li><strong>Rates of Change:<\/strong> Calculating instantaneous rates of change in practical contexts.<\/li>\n\n\n\n<li><strong>Kinematics:<\/strong> Modeling motion in a straight line, including displacement, velocity, and acceleration.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Context within Unit 3&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Topic 1:<\/strong> Covers Logarithmic Functions.<\/li>\n\n\n\n<li><strong>Topic 2:<\/strong> Further Differentiation and Applications (as detailed above).<\/li>\n\n\n\n<li><strong>Topic 3:<\/strong> Integration (reverse of differentiation).&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This unit is crucial as it forms a major part of internal assessment (IA2) and the external exam.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">********************************************************************************************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u306e\u6570\u5b66\u6307\u5c0e\u6cd5\u300c\u66f4\u306a\u308b\u5fae\u5206\u3068\u305d\u306e\u5fdc\u7528\u300d\uff08\u7279\u306b\u30e6\u30cb\u30c3\u30c83\u306e\u30c8\u30d4\u30c3\u30af2\uff09\u3067\u306f\u3001\u57fa\u672c\u7684\u306a\u591a\u9805\u5f0f\u5fae\u5206\u306b\u3068\u3069\u307e\u3089\u305a\u3001\u8d85\u8d8a\u95a2\u6570\uff08\u6307\u6570\u95a2\u6570\u3001\u5bfe\u6570\u95a2\u6570\u3001\u4e09\u89d2\u95a2\u6570\uff09\u3084\u9ad8\u5ea6\u306a\u898f\u5247\u3001\u305d\u3057\u3066\u305d\u308c\u3089\u306e\u5b9f\u8df5\u7684\u306a\u5fdc\u7528\u307e\u3067\u3001\u5fae\u7a4d\u5206\u6280\u6cd5\u3092\u62e1\u5f35\u3059\u308b\u3053\u3068\u306b\u91cd\u70b9\u3092\u7f6e\u3044\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30c8\u30d4\u30c3\u30af\u306e\u4e3b\u8981\u306a\u69cb\u6210\u8981\u7d20\u3092\u8a73\u7d30\u306b\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u5fae\u5206\u6280\u6cd5\uff08\u516c\u5f0f\u3068\u898f\u5247\uff09<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30bb\u30af\u30b7\u30e7\u30f3\u3067\u306f\u3001\u8907\u96d1\u306a\u5408\u6210\u95a2\u6570\u3092\u5fae\u5206\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u65b0\u3057\u3044\u5fae\u5206\u516c\u5f0f\u3068\u898f\u5247\u3092\u7fd2\u5f97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6307\u6570\u95a2\u6570\u306e\u5fae\u5206\uff1a(<math data-latex=\"e^{x}\"><semantics><msup><mi>e<\/mi><mi>x<\/mi><\/msup><annotation encoding=\"application\/x-tex\">e^{x}<\/annotation><\/semantics><\/math>) \u3068 (<math data-latex=\"e^{f(x)}\"><semantics><msup><mi>e<\/mi><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/msup><annotation encoding=\"application\/x-tex\">e^{f(x)}<\/annotation><\/semantics><\/math>) \u306e\u898f\u5247<br>\u5bfe\u6570\u95a2\u6570\u306e\u5fae\u5206\uff1a(<math data-latex=\"\\ln (x)\"><semantics><mrow><mrow><mi>ln<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\ln (x)<\/annotation><\/semantics><\/math>) \u3068 (<math data-latex=\"\\ln (f(x))\"><semantics><mrow><mrow><mi>ln<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\ln (f(x))<\/annotation><\/semantics><\/math>) \u306e\u898f\u5247\u3002\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206\uff1a(<math data-latex=\"\\sin (x)\"><semantics><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin (x)<\/annotation><\/semantics><\/math>)\u3001(<math data-latex=\"\\cos (x)\"><semantics><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cos (x)<\/annotation><\/semantics><\/math>)\u3001(<math data-latex=\"\\tan (x)\"><semantics><mrow><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\tan (x)<\/annotation><\/semantics><\/math>) \u306e\u898f\u5247\u3068\u305d\u308c\u3089\u306e\u9023\u9396\u5f8b\u306e\u7d44\u307f\u5408\u308f\u305b\u3002<br>\u9023\u9396\u5f8b\uff1a(<math data-latex=\"y=[f(x)]^{n}\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">[<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><msup><mo form=\"postfix\" stretchy=\"false\">]<\/mo><mi>n<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">y=[f(x)]^{n}<\/annotation><\/semantics><\/math>) \u306a\u3069\u306e\u5408\u6210\u95a2\u6570\u306b\u7528\u3044\u3089\u308c\u308b\u3002<br>\u7a4d\u5f8b\uff1a2\u3064\u306e\u95a2\u6570\u306e\u7a4d (<math data-latex=\"uv\"><semantics><mrow><mi>u<\/mi><mi>v<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">uv<\/annotation><\/semantics><\/math>) \u3092\u5fae\u5206\u3059\u308b\u306e\u306b\u7528\u3044\u3089\u308c\u308b\u3002<br>\u5546\u5f8b\uff1a2\u3064\u306e\u95a2\u6570\u306e\u5546 (<math data-latex=\"\\frac{u}{v}\"><semantics><mfrac><mi>u<\/mi><mi>v<\/mi><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{u}{v}<\/annotation><\/semantics><\/math>) \u3092\u5fae\u5206\u3059\u308b\u306e\u306b\u7528\u3044\u3089\u308c\u308b\u3002<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>\u5fae\u5206\u6cd5\u306e\u5fdc\u7528<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u308c\u306f\u3001\u4e0a\u8a18\u306e\u624b\u6cd5\u3092\u7528\u3044\u3066\u95a2\u6570\u3092\u5206\u6790\u3057\u3001\u73fe\u5b9f\u4e16\u754c\u306e\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u63a5\u7dda\u3068\u6cd5\u7dda\uff1a\u7279\u5b9a\u306e\u70b9\u306b\u304a\u3051\u308b\u66f2\u7dda\u306e\u63a5\u7dda\u3068\u6cd5\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b\u3002<br>\u505c\u7559\u70b9\uff1a\u4e00\u6b21\u5c0e\u95a2\u6570\u3068\u4e8c\u6b21\u5c0e\u95a2\u6570\u3092\u7528\u3044\u3066\u3001\u6700\u5927\u5024\u3001\u6700\u5c0f\u5024\u3001\u505c\u7559\u5909\u66f2\u70b9\u3092\u8b58\u5225\u3057\u3001\u5206\u985e\u3059\u308b\u3002<br>\u66f2\u7dda\u306e\u63cf\u753b\uff1a\u5c0e\u95a2\u6570\u3092\u7528\u3044\u3066\u3001\u5897\u52a0\uff0f\u6e1b\u5c11\u533a\u9593\u3001\u51f9\u72b6\uff08\u4e0a\u5411\u304d\uff0f\u4e0b\u5411\u304d\u306e\u51f9\u72b6\uff09\u3001\u304a\u3088\u3073\u5909\u66f2\u70b9\u3092\u6c42\u3081\u308b\u3002<br>\u6700\u9069\u5316\uff1a\u5b9f\u7528\u7684\u306a\u554f\u984c\u3092\u89e3\u304d\u3001\u6700\u5927\u5024\u307e\u305f\u306f\u6700\u5c0f\u5024\u3092\u6c42\u3081\u308b\uff08\u4f8b\uff1a\u5229\u76ca\u306e\u6700\u5927\u5316\u3001\u8868\u9762\u7a4d\u306e\u6700\u5c0f\u5316\uff09\u3002<br>\u5909\u5316\u7387\uff1a\u5b9f\u7528\u7684\u306a\u72b6\u6cc1\u306b\u304a\u3051\u308b\u77ac\u9593\u7684\u306a\u5909\u5316\u7387\u3092\u8a08\u7b97\u3059\u308b\u3002<br>\u904b\u52d5\u5b66\uff1a\u5909\u4f4d\u3001\u901f\u5ea6\u3001\u52a0\u901f\u5ea6\u3092\u542b\u3080\u76f4\u7dda\u904b\u52d5\u306e\u30e2\u30c7\u30eb\u5316\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30e6\u30cb\u30c3\u30c83\u306b\u304a\u3051\u308b\u6587\u8108<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30c8\u30d4\u30c3\u30af1\uff1a\u5bfe\u6570\u95a2\u6570\u3092\u6271\u3046\u3002<br>\u30c8\u30d4\u30c3\u30af2\uff1a\u3055\u3089\u306a\u308b\u5fae\u5206\u5316\u3068\u5fdc\u7528\uff08\u4e0a\u8a18\u53c2\u7167\uff09\u3002<br>\u30c8\u30d4\u30c3\u30af3\uff1a\u7a4d\u5206\uff08\u5fae\u5206\u306e\u9006\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30e6\u30cb\u30c3\u30c8\u306f\u3001\u5185\u90e8\u8a55\u4fa1\uff08IA2\uff09\u3068\u5916\u90e8\u8a66\u9a13\u306e\u4e3b\u8981\u90e8\u5206\u3092\u5360\u3081\u308b\u305f\u3081\u3001\u975e\u5e38\u306b\u91cd\u8981\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In QLD Year 12 Mathematical Methods, &#8220;Further Differentiation and Applications&#8221; (specifically Topic 2 of Unit 3) focuses on expanding calculus techniques beyond basic polynomial differentiation to include transcendental functions (exponential, logarithmic, trigonometric) and advanced rules, alongside their practical applications.&nbsp; Here is a detailed breakdown of the key components of this topic:&nbsp; 1. Differentiation Techniques (Formulas [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1340","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1340","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1340"}],"version-history":[{"count":3,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1340\/revisions"}],"predecessor-version":[{"id":1377,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1340\/revisions\/1377"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1340"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1340"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1340"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}