You Should Know
\cdot 3x^2 = 18x^2 </p><ul><li><p><strong>LogarithmicRule:</strong><br></p></li></ul><p></p><ul><li><p>Example:<br></p></li></ul><p></p><h4id="d7228863−1611−41c7−8b18−27a290f4bb67"data−toc−id="d7228863−1611−41c7−8b18−27a290f4bb67"collapsed="false"seolevelmigrated="true">ChallengeswithDerivativesofComplicatedFunctions</h4><ul><li><p>Thegoalistoexpressthederivativeof“complicated”functionsintermsofthepiecesthatmakeupthe“complicated”functions.</p></li><li><p>Forinstance,thederivativesofthefunctions</p></li></ul><p></p><p></p><ul><li><p>areknown,butcomplicationsarisewithproductsoffunctionslike</p></li></ul><p></p><p></p><ul><li><p>Unfortunately,derivativesarenotasfriendlywithmultiplicationanddivisionbyvariables!</p></li></ul><h3id="1b0b3bc1−1f60−4602−996e−a8f74db09561"data−toc−id="1b0b3bc1−1f60−4602−996e−a8f74db09561"collapsed="false"seolevelmigrated="true">ThePRODUCTRule</h3><p><br></p><ul><li><p>Examples:</p></li></ul><p></p><p></p><p></p><p></p><h3id="3bb27fc3−ad03−412c−a69c−6335c1c943a2"data−toc−id="3bb27fc3−ad03−412c−a69c−6335c1c943a2"collapsed="false"seolevelmigrated="true">TheQUOTIENTRule</h3><p><br></p><ul><li><p>Examples:</p></li></ul><p></p><p></p><p></p><p></p><h3id="3abb04f1−231e−44be−9556−c248a958cf75"data−toc−id="3abb04f1−231e−44be−9556−c248a958cf75"collapsed="false"seolevelmigrated="true">FunctionCompositionandChainRule</h3><h4id="710a7b79−54c9−4278−aa0b−908c5e601202"data−toc−id="710a7b79−54c9−4278−aa0b−908c5e601202"collapsed="false"seolevelmigrated="true">ReminderonFunctionComposition</h4><ul><li><p>Anunderstandingoffunctioncompositionisvital:</p><ul><li><p>Forfunctions</p></li><li><p> f(x) = x^2 + x + 1 </p></li><li><p>g(x)=e^{x}</p></li><li><p> h(x) = x </p></li><li><p>Thefunctioncompositioncantakeformsuchas:</p></li><li><p> (g \circ f)(x) = g(f(x)) </p></li><li><p>Forexample,calculatedas:<br> g(f(x)) = e^{x^2 + x + 1} </p></li></ul></li></ul><h4id="21951d2f−e2e9−4883−9110−62b399c829fd"data−toc−id="21951d2f−e2e9−4883−9110−62b399c829fd"collapsed="false"seolevelmigrated="true">ChainRuleforDifferentiation</h4><ul><li><p>TheChainRuleallowsdifferentiationofcomposedfunctions:</p><ul><li><p>Thechainruleisexpressedas:<br> \frac{d}{dx}(f(g(x))) = f'(g(x))g'(x) </p></li><li><p>Thedifferentiationisperformedbydifferentiatingtheouterfunctionwhilekeepingtheinnerfunctionintact,followedbymultiplyingbythederivativeoftheinnerfunction.</p></li></ul></li></ul><h4id="41f6e5ad−a1fb−4d21−b4ef−3e3d56d5e85c"data−toc−id="41f6e5ad−a1fb−4d21−b4ef−3e3d56d5e85c"collapsed="false"seolevelmigrated="true">ChainRuleExamples</h4><ul><li><p>Someexamplesinclude:</p><ul><li><p>Simpleapplicationofthechainrule:</p></li><li><p> \frac{d}{dx} \left( e^{6x^2 + 3x - 4} \right) </p></li><li><p>Furtherapplicationsassumingexplicitfunctionswithderivativesnoted.</p></li></ul></li></ul><h3id="4747d7fb−0f00−4990−bec2−c2bad4b7036b"data−toc−id="4747d7fb−0f00−4990−bec2−c2bad4b7036b"collapsed="false"seolevelmigrated="true">SummaryofDerivativeRules</h3><ul><li><p>Tosummarizeessentialderivativerules:</p><ul><li><p><strong>PowerRule:</strong> \frac{d}{dx}(x^n) = n x^{n-1} </p></li><li><p><strong>ExponentialRule:</strong> \frac{d}{dx}(e^{kx}) = ke^{kx} </p></li><li><p><strong>LogarithmicRule:</strong> \frac{d}{dx} \ln(x) = \frac{1}{x} </p></li><li><p><strong>ProductRule:</strong> \frac{d}{dx}(f(x)g(x)) = f(x)g'(x) + f'(x)g(x) </p></li><li><p><strong>QuotientRule:</strong> \frac{d}{dx}(\frac{f(x)}{g(x)}) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2} </p></li><li><p><strong>ChainRule:</strong> \frac{d}{dx}(g(f(x))) = g'(f(x))f'(x) $$
Homework and Application
-Homework 7 includes problems for practice that will be demonstrated in lecture videos, depending on time constraints.