13notes.pdf - Section 3.6 The Chain Rule Math 21A Spring 2020 UC Davis What Now What Next 3.4 3.5 3.6 3.7 3.8 3.9 3.10 The Derivative as a Rate of

13notes.pdf - Section 3.6 The Chain Rule Math 21A Spring...

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Section 3.6: The Chain Rule Math 21A, Spring 2020, UC Davis
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What Now, What Next? 3.4 The Derivative as a Rate of Change 3.5 Derivatives of Trigonometric Functions 3.6 The Chain Rule 3.7 Implicit Differentiation 3.8 Derivatives of Inverse Functions and Logarithms 3.9 Inverse Trigonometric Functions 3.10 Related Rates · · ·
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The Problem – Derivatives of Compositions How is ( f g ) 0 related to f 0 and g 0 ? Example: What is the derivative of sin( x 2 )?
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The Problem – Derivatives of Compositions How is ( f g ) 0 related to f 0 and g 0 ? Example: What is the derivative of sin( x 2 )? Solution: Given a composition y = f ( g ( x )) we write u = g ( x ), so that y = f ( u ). Then we have dy dx = dy du · du dx . Example: Given y = sin( x 2 ) we write y = sin( u ) with u = x 2 . Then dy dx = dy du · du dx = cos( u ) · 2 x = cos( x 2 ) · 2 x .
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Derivation: The Case of Linear Functions Silly Example: Let y = 3 u + 5 and u = 1 2 x - 1, so that y = 3 ( 1 2 x - 1 ) + 5 = 3 2 x + 2 . Then clearly dy dx = 3 2 . Note that we simply get this by multiplying the slopes of the two linear functions: dy dx = 3 2 = 3 · 1 2 = dy du · du dx .
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