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Math 140

7.2 – Partial Derivatives

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A ________________ is the derivative of a function in two (or more) variables with respect to one variable, while holding all other variables _________.

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A ________________ is the derivative of a function in two (or more) variables with respect to one variable, while holding all other variables _________.

partial derivative

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A ________________ is the derivative of a function in two (or more) variables with respect to one variable, while holding all other variables _________.

partial derivative

constant

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Ex 1.Find the partial derivative of with respect to (holding constant).

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Notation:Suppose .

or means the partial derivative of with respect to (holding constant).

or means the partial derivative of with respect to (holding constant).

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Ex 2.Find and if .

Ex 3.Find the partial derivatives and if .

Ex 4.Find the partial derivatives and if .

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There are four __________________________:

Note: and are called ______________.

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There are four __________________________:

Note: and are called ______________.

second-order partial derivatives

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There are four __________________________:

Note: and are called ______________.

second-order partial derivatives

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There are four __________________________:

Note: and are called ______________.

second-order partial derivatives

mixed partials

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Ex 5.Compute the four second-order partial derivatives of .

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Sometimes a product can affect another product’s demand. For example, if more cans of Coke are sold, then less bottles of water are likely to be sold. So, Coke and water are said to be _____________________. If more hotdogs are sold, then more hotdog buns are likely to be sold. So, hotdogs and hotdog buns are said to be ___________________________.

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Sometimes a product can affect another product’s demand. For example, if more cans of Coke are sold, then less bottles of water are likely to be sold. So, Coke and water are said to be _____________________. If more hotdogs are sold, then more hotdog buns are likely to be sold. So, hotdogs and hotdog buns are said to be ___________________________.

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Sometimes a product can affect another product’s demand. For example, if more cans of Coke are sold, then less bottles of water are likely to be sold. So, Coke and water are said to be _____________________. If more hotdogs are sold, then more hotdog buns are likely to be sold. So, hotdogs and hotdog buns are said to be ___________________________.

substitute commodities

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Sometimes a product can affect another product’s demand. For example, if more cans of Coke are sold, then less bottles of water are likely to be sold. So, Coke and water are said to be _____________________. If more hotdogs are sold, then more hotdog buns are likely to be sold. So, hotdogs and hotdog buns are said to be ___________________________.

substitute commodities

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Sometimes a product can affect another product’s demand. For example, if more cans of Coke are sold, then less bottles of water are likely to be sold. So, Coke and water are said to be _____________________. If more hotdogs are sold, then more hotdog buns are likely to be sold. So, hotdogs and hotdog buns are said to be ___________________________.

substitute commodities

complementary commodities

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Ex 6.Suppose the demand function for hotdogs is given by and the demand for hotdog buns is given by . Here, is the dollar price for a hotdog, and is the dollar price for a hotdog bun. Determine whether hotdog and hotdog buns are substitute commodities, complementary commodities, or neither.