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Calculus 1000A/B Lecture Notes - Lecture 21: Function Composition, Product Rule

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18 Oct 2018

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Calculus 1000A/B Lecture Notes - Lecture 20: Chain Rule, Power Rule, The Thirteen Chairs

Calculus 1000A/B Lecture Notes - Lecture 21: Function Composition, Product Rule

Calculus 1000A/B Lecture 22: Calculus 1000 A -Lecture 21- Section 3.5 Implicit Differentiation

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Calculus 1000a lecture 21 - section 3. 4 (continued) the chain rule. Quiz wednesday next week; sample quiz and sample midterm exam to be posted; quiz 2 available for pickup. Deriving the formula of the derivative of the exponential function (cid:1856)(cid:1877)(cid:1856)(cid:1876)=(cid:1856)(cid:1877)(cid:1856)(cid:1873) (cid:1856)(cid:1873)(cid:1856)(cid:1876) Recall section 3. 1 (independent review: (cid:1858)(cid:4666)(cid:1876)(cid:4667)=(cid:3051),>(cid:882) background knowledge. Example 1: (cid:1877)=(cid:1857) (cid:3051: remember, we can solve this by introducing a new variable, (cid:1877)=(cid:1857) (cid:3051)= (cid:1857)(cid:3048), (cid:1875) (cid:1857)(cid:1857) (cid:1873)= (cid:1876, (cid:3031)(cid:3052)(cid:3031)(cid:3051)=(cid:3031)(cid:4666)(cid:3032)(cid:3296)(cid:4667)(cid:3031)(cid:3048) (cid:3031)(cid:4666) (cid:3051)(cid:4667)(cid:3031)(cid:3051) =(cid:1857)(cid:3048) (cid:883)= (cid:1857) (cid:3051) In the last step above, we again replace the u in terms of x. We might see examples where there is more than one inner function. In this instance: (cid:4666)(cid:1876)(cid:4667)=(cid:1858)(cid:4666)(cid:1859)( (cid:4666)(cid:1876)(cid:4667))(cid:4667): if f, g and h are differentiable, then f is differentiable and the derivative f": (cid:1856)(cid:1877)(cid:1856)(cid:1876)=(cid:1856)(cid:1877)(cid:1856)(cid:1873) (cid:1856)(cid:1873)(cid:1856)(cid:1874) (cid:1856)(cid:1874)(cid:1856)(cid:1876) Example 3: we might have an example where information is given to us, but no complete equation if u= g(h(x)) and is given: given: h(x)= f(x)(g(x)) and f(x)=f(g(x)).

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