partial derivatives - x etc 3 Second Order Partial...

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12.3 Partial Derivatives 1-Variable Functions The derivative of f ( x ) at x = a is f ( a ) = df dx x = a = Multivariable Functions Defn. Partial Derivative The partial derivative of f ( x, y ) with respect to x at the point ( a, b ) is f x ( a, b ) = ∂f ∂x ( a,b ) = provided the limit exists. Similarly, the partial derivative of f ( x, y ) with respect to y at the point ( a, b ) is f y ( a, b ) = ∂f ∂y ( a,b ) = provided the limit exists. 1
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Geometric Interpretation 1-Variable 2-Variable ∂f ∂x = ∂f ∂y = As in the 1-variable case, we can compute f x , f y algebraically. Example 1 Compute the partials of f ( x, y ) = 1 + x + y - 3 x 2 y at (2 , 3) 2
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Example 2 f ( x, y ) = 300 e x 2 +2 y 2 Example 3 f ( r, m ) = m r + m Example 4 g ( x, y ) = tan 2 (3 x 2 + 5 y ) Know product/quotient rule, chain rule, and basic derivatives (trig, ln
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Unformatted text preview: x , etc.) 3 Second Order Partial Derivatives f xx = f yy = f xy = f yx = Example 1 f ( x, y ) = cos( xy ) Note: f xy = f yx for functions such that f xx , f yy , f xy , and f yx all exist and are continuous. 4 Example 2 z = 1-x + y-3 x 2 y Theorem. Differentiability of f ( x, y ) If the partial derivatives f x and f y exist near ( a, b ) and are continuous at ( a, b ), then f is differentiable at ( a, b ). Geometric Interpretation f xx = f yy = f xy = f yx = 5...
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