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Unformatted text preview: The Chain Rule
The Chain Rule says that if , then Sometimes we say: Given and , then So, in Here we decompose the function into two functions and , which are then, with we have and and so that 2. decompose some function, where some function , so that then where MTH173 section 3.5 notes page 1 8. what is here 14. cos cos cos cos 20. cos cos sin sin for the second factor, we have so that, using the product rule on the given function, we have MTH173 section 3.5 notes page 2 30. sin cos here we must use the quotient rule, so for starters we have sin for sin cos cos , we must use the chain rule: where sin , so cos sin cos sin cos Then we can complete the derivative of the function givenin the problem: sin cos cos cos sin cos This result can be "simplified" An alternative for this one would be to rewrite the original function: sin cos sin cos sin tan sin and then just use the product rule 34. sin here we use the product rule: sin sin but then sin for sin cos needs the chain rule: sin where cos above, giving:
MTH173 section 3.5 notes page 3 , so that and then we substitute in sin cos 44. eqn tangent to sin sin at since it is the equation of a line thru the origin. : we know that the equation will start as Then is the value of the derivative evaluated at cos cos line is sin cos sin cos or, usually, just . , so the equation of the tangent Graph to check: MTH173 section 3.5 notes page 4 36. Here tan is a constant. is , so here we have The derivative of and then the chain rule gives . So now we have tan sec so that we substitute in
tan tan , so that which also requires the chain rule with sec sec MTH173 section 3.5 notes page 5 56. if , what is where ? Another way: if so , what is , where , so or ! ! " " " " " ! in then where and . 64. Let # Determine # # but $ and so we substitute to obtain # MTH173 section 3.5 notes page 6 ...
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