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Engineering Calculus Notes 274

# Engineering Calculus Notes 274 - 262 CHAPTER 3 REAL-VALUED...

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262 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION We will exploit this equation in two ways. For notational convenience, we will drop reference to where a partial is being taken: for the rest of this proof , ∂f ∂x = ( x,y ) ∂y = ( ) where −→ x = ( ) is the point at which we are trying to prove di±erentiability of φ . Moving the ²rst two terms to the left side, dividing by ( x )( ∂f ∂y ), and taking absolute values, we have v v v v y x + ∂f/∂x ∂f/∂y v v v v = | ε | | | b ( x, y ) b |△ x | | ε | | | b 1 + v v v v y x v v v v B (3.20) (since b ( x, y ) b ≤ |△ x | + |△ y | ). To complete the proof, we need to ²nd an upper bound for v v v 1 + y x v v v on the right side. To this end, we come back to Equation ( 3.19 ), this time moving just the second term to the left, and then take absolute values, using the triangle
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