Econ501aL1 - Functions of Several Variables: Partial...

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Functions of Several Variables: Partial Derivatives Consider the function, z = f ( x; y ). Now there is no such thing as \the" derivative of z. It depends which direction you are moving in. -4 -2 0 2 4 x -4 -2 0 2 4 y -50 -40 -30 -20 -10 0 The partial derivative of z with respect to x , @z @x ,isthe slope of the function in the direction of the x-axis. In other words, it is the rate at which z changes in response to a small increase in x, holding y constant. This is sometimes written as f x ( x; y )o r @f ( x;y ) @x .W e
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To compute partial derivatives, treat all the other vari- ab lesasconstants . example: z = x 2 y 2 +3 xy @z @x =2 xy 2 +3 y and @z @y =2 x 2 y +3 x: Second partials: @ 2 z @x 2 =2 y 2 , @ 2 z @y 2 =2 x 2 ,and @ 2 z @x@y = 4 xy
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To use the chain rule , the total e®ect on a function of several variables is the sum of the e®ects from each variable separately. Thus, if we have z ( ® )= f
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Econ501aL1 - Functions of Several Variables: Partial...

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