notes-partials1 - Multivariable Calculus Partial...

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ECON 331 Multivariable Calculus Partial Derivatives Single variable calculus is really just a ”special case” of multivariable calculus. For the function y = f ( x ) , we assumed that y was the endogenous variable, x was the exogenous variable and everything else was a parameter. For example, given the equations y = a + bx or y = ax n we automatically treated a , b ,and n as constants and took the derivative of y with respect to x ( dy/dx ). However, what if we decided to treat x as a constant and take the derivative with respect to one of the other variables? Nothing precludes us from doing this. Consider the equation y = ax where dy dx = a Now suppose we f nd the derivative of y with respect to a , but TREAT x as the constant. Then dy da = x Here we just ”reversed” the roles played by a and x in our equation. Two Variable Case: let z = f ( x, y ) ,wh ichmeans z is a function of x and y .Inth i sc a s e z is the endoge- nous (dependent) variable and both x and y are the exogenous (independent) variables. To measure the the e f ect of a change in a single independent variable (x or y) on the dependent variable (z) we use what is known as the PARTIAL DERIVATIVE . The partial derivative of z with respect to x measures the instantaneous change in the function as x changes while HOLDING y constant . Similarly, we would hold x constant if we wanted to evaluate the e f ect of a change in y on z. Formally: z x is the ”partial derivative” of z with respect to x ,t r e a t in g y as a constant. Sometimes written as f x . z y is the ”partial derivative”
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The symbol (”bent over” lower case D) is called the ”partial” symbol. It is interpreted in exactly the same way as dy dx from single variable calculus. The symbol simply serves to remind us that there are other variables in the equation, but for the purposes of the current exercise, these other variables are held constant. EXAMPLES:
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This note was uploaded on 09/13/2009 for the course EOCNOMICS Econ 331 taught by Professor Kelvinkwainger during the Spring '09 term at Simon Fraser.

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notes-partials1 - Multivariable Calculus Partial...

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