PARTIAL DERIVATIVES - Copy (2)

# PARTIAL DERIVATIVES - Copy (2) - A Alaca MATH 1005 Winter...

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A. Alaca MATH 1005 Winter 2010 1 MATH 1005 WINTER 2010 LECTURE SLIDES Prepared by Ay¸ se Alaca Last modifed: January 1, 2010 These Slides replace neither the Text Book nor the Lectures PARTIAL DERIVATIVES

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A. Alaca MATH 1005 Winter 2010 2 Partial Derivatives of Two Variable Functions If y = f ( x ), then f ± ( x ) = lim h 0 f ( x + h ) - f ( x ) h . If z = f ( x, y ), then ∂z ∂x = lim h 0 f ( x + h, y ) - f ( x, y ) h , ∂z ∂y = lim h 0 f ( x, y + h ) - f ( x, y ) h Notation: ∂z ∂x = ∂f ∂x = f x ( x, y ), ∂z ∂y = ∂f ∂y = f y ( x, y ). The partial derivatives f x ( x, y ) and f y ( x, y ) are called the ±rst order partial derivatives of f ( x, y ). Example: Let z = f ( x, y )= x 2 y . ∂z ∂x = lim h 0 ( x + h ) 2 y - x 2 y h = lim h 0 2 xhy + h 2 y h = lim h 0 (2 xy + hy )=2 xy. ∂z ∂y = lim h 0 x 2 ( y + h ) - x 2 y h = lim h 0 x 2 h h = lim h 0 x 2 = x 2 . Note:
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## This note was uploaded on 03/08/2010 for the course MATH 1005 taught by Professor Any during the Winter '07 term at Carleton CA.

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PARTIAL DERIVATIVES - Copy (2) - A Alaca MATH 1005 Winter...

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