Engineering Calculus Notes 270

Engineering Calculus Notes 270 - 3.16 for y in terms of x...

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258 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION (3 , 2) Figure 3.3: The Curve x 3 + y 3 + xy = 13 A clue to what is going on can be found by recasting the process of implicit di±erentiation in terms of level curves. Suppose that near the point ( x 0 ,y 0 ) on the level set L ( f,c ) f ( x,y ) = c (3.16) we can (in principle) solve for y in terms of x : y = φ ( x ) . Then the graph of this function can be parametrized as −→ p ( x ) = ( x,φ ( x )) . Since this function is a solution of Equation (
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Unformatted text preview: 3.16 ) for y in terms of x , its graph lies on L ( f,c ): f ( −→ p ( x )) = f ( x,φ ( x )) = c. Applying the Chain Rule to the composition, we can di±erentiate both sides of this to get ∂f ∂x + ∂f ∂y dy dx = 0 and, provided the derivative ∂f/∂y is not zero , we can solve this for φ ′ ( x ) = dy/dx : φ ′ ( x ) = dy dx = − ∂f/∂x ∂f/∂y ....
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