Calculus II Notes 15.3 - Calculus II- Stewart Dr. Berg...

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Unformatted text preview: Calculus II- Stewart Dr. Berg Spring 2010 Page 1 15.3 15.3 Partial Derivatives The derivative of a function of one variable is simple to define and has a simple interpretation. Functions of more than variable present a greater challenge. A useful and simple beginning is the partial derivative. Functions of Two Variables Definition Let f be a function of two variables x and y . The its partial derivatives are: f x ( x , y ) = lim h → f ( x + h , y ) − f ( x , y ) h and f y ( x , y ) = lim h → f ( x , y + h ) − f ( x , y ) h . Example A Let f ( x , y ) = x 2 + xy + y 2 . Then f x ( x , y ) = lim h → f ( x + h , y ) − f ( x , y ) h = lim h → x + h ( ) 2 + ( x + h ) y + y 2 [ ] − x 2 + xy + y 2 [ ] h = lim h → x 2 + 2 xh + h 2 + xy + hy + y 2 − x 2 − xy − y 2 h = lim h → 2 xh + h 2 + hy h = lim h → 2 x + h + y ( ) = 2 x + y Similarly f y ( x , y ) = x + 2 y . Notation : If z = f ( x , y ) , then f x ( x , y ) = f x = ∂ f ∂ x = ∂ ∂ x f ( x , y ) = ∂ z ∂ x = f 1 = D 1 f = D x f and f y ( x , y ) = f y = ∂ f ∂ y = ∂ ∂ y f ( x , y ) = ∂ z ∂ y = f 2 = D 2 f = D y f ....
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This note was uploaded on 04/04/2011 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas at Austin.

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Calculus II Notes 15.3 - Calculus II- Stewart Dr. Berg...

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