handout 14-3 - Math 200, Spring 2010 Handout 11 Section...

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Math 200, Spring 2010 Handout 11 Section 14-3 Partial Derivatives Partial Derivatives The partial derivatives are the rates of change with respect to each variable separately. A function ( ) , f x y in two variables has two partial derivatives, denoted x f and y f , defined by the following limits (if they exist): ( ) ( ) ( ) 0 , , , lim x h f a h b f a b f a b h + = , which is the partial derivative of f with respect to x at ( ) , a b and ( ) ( ) ( ) 0 , , , lim y h f a b h f a b f a b h + = , which is the partial derivative of f with respect to y at ( ) , a b Thus, ( ) , x f a b is the obtained by keeping y fixed ( ) y b = and finding the ordinary derivative at a of the function ( ) ( ) , g x f x b = , and ( ) , y f a b is the obtained by keeping x fixed ( ) x a = and finding the ordinary derivative at b of the function ( ) ( ) , G y f a y = . Rule for Finding Partial Derivatives of z = f ( x , y ) 1. To find x f , regard y as a constant and differentiate ( ) , f x y with respect to x . 2. To find y f , regard x as a constant and differentiate ( ) , f x y with respect to y . 1. Find the first partial derivatives of the function ( ) 4 3 2 , 8 f x y x y x y = + . 2. Find the partial derivative ( ) 2,1 x f for the function ( ) 4 3 2 , 8 f x y x y x y = + . "otations for Partial Derivatives
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handout 14-3 - Math 200, Spring 2010 Handout 11 Section...

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