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way."},"version":77,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"29f8a483928726bc","login":"axiom7","name":"Axiom","avatar_url":"https://avatars-next.observableusercontent.com/ad1f41069aec7011f5912c322e8999450aaf31252cd1d9a43f1ed7551f2c4ea1","tier":null,"bio":""}],"members":[{"type":"individual","id":"29f8a483928726bc","login":"axiom7","name":"Axiom","avatar_url":"https://avatars-next.observableusercontent.com/ad1f41069aec7011f5912c322e8999450aaf31252cd1d9a43f1ed7551f2c4ea1","tier":null,"role":"editor"},{"type":"team","id":"09c31aeb403ca87c","login":"axiom-tutor","name":"AxiomTutor","avatar_url":"https://avatars-next.observableusercontent.com/2bf1d793dac61ef22bdbbceae725f52d3b12334d88b2ecaf61864a154e830f64","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Exercises for chapters 2.3 through 2.5","cells":[{"id":0,"value":"# Exercises for chapters 2.3 through 2.5\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":7,"value":"## Products, Quotients, and Trig Rules\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":26,"value":"### Book Exercises\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":30,"value":"

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":33,"value":" $\"image@1.png\"$ ","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":37,"value":"

Many classes skip the following kind of problem, but some of the harder calculus classes will expect you to solve it. Feel free to solve it if you think you might need to.

\n\n $\"image@2.png\"$ ","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":9,"value":"### Exercise 1\n\nFind ${tex`\\frac d {dx}\\left(\\frac {x^2+x}{x}\\right)`}.\n\nAlso find ${tex`\\frac{d}{dx} \\left(\\frac {x}{x^2+x}\\right)`}.\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":13,"value":"### Exercise 2\n\nFind ${tex`\\left(\\frac{\\cos x}{\\sin x}\\right)'`}.\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":16,"value":"### Exercise 3\n\nFind ${tex`[(\\cos x)(\\cos x)]'`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":20,"value":"### Exercise 4\n\nLet ${tex`f,g,h`} be functions. Find ${tex`(fgh)'`}. \n\nYour answer should contain products and sums of ${tex`f, g, h, f', g', h'`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":23,"value":"## Chain Rule\n\nThe part of the chain rule that most people get wrong is that they fail to understand how functions decompose. First let's make sure you understand composition of functions.\n\nFor example, if ${tex`f(x) = \\sqrt{x}`} and ${tex`g(x) = 2x-1`} then here is their composition: \n\n${tex.block`(f\\circ g)(x) = \\sqrt{2x-1}`}\n\n### Exercise 1\n\nFor ${tex`f(x) = \\sqrt{x}`} and ${tex`g(x) = 2x-1`} find the composition ${tex`(g\\circ f)(x)`}.\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":44,"value":"### Exercise 2\n\nFor ${tex`e^{x^2}`} this function can be described as the composition of ${tex`f(x) = e^x`} with ${tex`g(x) = x^2`}. This is called the \"decomposition\" of the original function into ${tex`f`} and ${tex`g`}.\n\nNow find the decomposition of ${tex`e^{\\cos x}`}.\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":47,"value":"### Exercise 3\n\nFinding the decomposition of a function is the first step of using the Chain Rule. Consider ${tex`\\cos(x^2)`} which is the composition of ${tex`f(x)=\\cos(x)`} with ${tex`g(x)=x^2`}. Therefore if we want the derivative of ${tex`\\cos (x^2)`} then we want ${tex`\\frac{d}{dx} (f\\circ g)(x)`}.\n\nThe chain rule says that ${tex`\\frac d {dx}(f\\circ g)(x) = (f'\\circ g)(x) \\ \\cdot \\ g'(x)`}.\n\n#### Part 1\n\nSo first find ${tex`f'`} and ${tex`g'`}.\n\n#### Part 2\n\nNext find ${tex`f'\\circ g`}.\n\n#### Part 3\n\nUse what you found in parts 1 and 2 to find ${tex`\\frac{d}{dx}(f\\circ g)(x)`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":53,"value":"### Exercise 4\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":57,"value":"Now do it from scratch, where I don't tell you the decomposition. \n\n#### Part 1\n\nFind the decomposition of ${tex`(\\cos x)^2`}.\n\n#### Part 2\n\nUse the decomposition and the chain rule to find ${tex`\\frac{d}{dx}(\\cos x)^2`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":59,"value":"### Exercise 5\n\nSometimes a function's decomposition isn't so easy. Find the decomposition of \n\n${tex.block`(\\sqrt{x+1}+1)^2`}\n\nand use it to find the derivative of this function. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":62,"value":"### Book Exercises\n\nFeel free to do as many of these as you like:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":64,"value":" $\"image@3.png\"$ ","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":70,"value":" $\"image@4.png\"$ ","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":72,"value":" $\"image@5.png\"$ ","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":74,"value":" $\"image@6.png\"$ ","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":76,"value":"## Implicit Differentiation\n\nIf you have extra time, feel free to read up on implicit differentiation and ask questions about it during Friday's meeting.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"}],"files":"$W31","stdlib":"1"}} 30:"$2f:body"