Engineering Calculus Notes 260

Engineering Calculus Notes 260 - 248 CHAPTER 3. REAL-VALUED...

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Unformatted text preview: 248 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION or √ ∂f 3 =√ ∂ρ 22 √ 3 ∂f =√ ∂φ 2 1 ∂f =√ . ∂θ 2 The value and derivative of g(y ) at y = f 2, π , π = 43 1 g√ 2 1 g′ √ 2 1 √ 2 are 1 = − ln 2; 2 √ =2 and from this we get 1 ∂z ∂f = g′ √ ∂ρ 2 ∂ρ √ √ 3 √ =2 22 √ 3 = 2 ∂z 1 ∂f = g′ √ ∂φ 2 ∂φ √ √ 3 =2√ 2 √ =3 ∂z 1 ∂f = g′ √ ∂θ 2 ∂θ √ 1 =2√ 2 = 1. Note that this formula could have been found directly, using Definition 3.3.3 (Exercise 7): the substantive part of the proof above was to show that the composite function is differentiable. ...
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