N y fx x x a common example used in economics is the

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n y = f(x , x , ... , x ) ! A common example used in economics is the Cobb-Douglas production function where Y represents the output produced by a firm, or GDP if a country is being considered, K represents capital, L represents labour, and A , " and $ are “parameters” or constants: Y = A K L " $ e e e e or log Y = log A + " log K + $ log L 25. Partial Differentiation ! We differentiate functions of several variables with respect to each of their independent variables by assuming that the other variables remain constant . ! This is called "partial" differentiation as opposed to “ordinary” differentiation for functions of one variable. ! Thus, y = f (x, z) has two partial derivatives: (i) the partial derivative of y with respect to x , written as: "partial dee y by dee x " etc. (ii) the partial derivative of y with respect to z , written as: "partial dee y by dee z " etc. ! Note that the Greek letter M (delta) is used to distinguish partial derivatives from ordinary, one-variable derivatives for which we use the letter d . 1 2 n ! When the notation y = f ( x , x , ..., x ) is used, the partial derivatives can be written as:
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68 1 1 2 n 1 f (x , x , ..., x ) - partial derivative with respect to x 2 1 2 n 2 f (x , x , ..., x ) - partial derivative with respect to x 3 1 2 n 3 f (x , x , ..., x ) - partial derivative with respect to x etc. Examples : (i) Recall the aggregate consumption function in section 21.1: C = f (Y) = " + $ Y where Y represents total income and is the marginal propensity to consume .
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