L13 - Lecture 13 Consequences of Chain Rule Implicit...

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Lecture 13 — Consequences of Chain Rule; Implicit Differentiation Given an identity, such as sin 2 ( x ) + cos 2 ( x ) = 1 for all real x , we can think of it as an assertion that two functions are the same, and thus should have the same derivative: R ( x ) = sin 2 ( x ) + cos 2 ( x ) S ( x ) = 1 Now, call f ( x ) = cos( x ) . We can find the formula for its derivative (we omitted the proof) from the fact that it shares this relationship with sin( x ) . We have: (1) (2) We can now SOLVE for df dx to see its formula!
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Note that equation (2) follows from (1) for ANY dif- ferentiable function x ( θ ) , since by chain rule: d dx [ f ( x )] 2 = 2 [ f ( x )] df dx So, if we had not known the formula for f ( x ) , it was good enough to observe that it satisfied an identity involving sin( x ) and 1 , two functions with known derivatives. Suppose we have the identity [ P ( t ) ] 3 + P ( t ) = t 4 + 1 for all times t . What must be the derivative of P with respect to t ?
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This note was uploaded on 05/27/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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L13 - Lecture 13 Consequences of Chain Rule Implicit...

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