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# slides0-handout - Econ407 J Tessada Spring 2009 Intro...

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Econ407 J. Tessada Spring 2009 Intro Derivatives Growth Rates Optimization Constrained Optimization Econ407 Advanced Macroeconomics Lecture 0: A Quick Math Refresher Jos´e Tessada University of Maryland - College Park January 29, 2009 Econ407 J. Tessada Spring 2009 Intro Derivatives Growth Rates Optimization Constrained Optimization Announcements Web: http://sites.google.com/site/econ407 Syllabus updated with some links to readings Next two lectures: We will review what the evidence tells us about economic growth No readings assigned After that we will start studying the Solow growth model

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Econ407 J. Tessada Spring 2009 Intro Derivatives Simple Derivatives Partial Derivatives Growth Rates Optimization Constrained Optimization Derivatives: A Refresher The derivative of a function y = f ( x ) at the point P = ( x 1 , f ( x 1 )) is the slope of the tangent line at that point The derivative is also a “function” The derivative of a function represents the ratio of the change in the function when the argument variables changes an inﬁnitesimal amount Let y = f ( x ) be a function, then the derivative of y with respect to x , f 0 ( x ) is deﬁned as f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h if the limit exists and is ﬁnite. Econ407 J. Tessada Spring 2009 Intro Derivatives Simple Derivatives Partial Derivatives Growth Rates Optimization Constrained Optimization Derivatives: Some Notation A derivative can be denoted in many different ways Let us again use the example of y = f ( x ) , its derivative can be written as any of the following f 0 ( x ) y 0 dy dx df dx d dx [ f ( x )] If we evaluate the derivative at a particular point x = a we write f 0 ( a ) or dy dx ± ± ± ± a Note: not every continuous function is differentiable (i.e., they are not the same concept)
Econ407 J. Tessada Spring 2009 Intro Derivatives Simple Derivatives Partial Derivatives Growth Rates Optimization Constrained Optimization Rules of Differentiation I Some usual rules ( a , b , n are constants), all are derivatives with respect to x Constant functions: f ( x ) = a f 0 ( x ) = 0 Additive and multiplicative functions f ( x ) = a + bx f 0 ( x ) = b Powers of x f ( x ) = x n f 0 ( x ) = nx n - 1 Natural logs f ( x ) = ln ( x ) f 0 ( x ) = 1/ x Exponential f ( x ) = e ax f 0 ( x ) = ae ax where e is the exponential function Functions of the form a x f ( x ) = a x f 0 ( x ) = a x ln ( a ) Econ407 J. Tessada Spring 2009 Intro Derivatives Simple Derivatives Partial Derivatives Growth Rates Optimization Constrained Optimization Rules of Differentiation II Sums f ( x ) = g ( x ) + h ( x ) f 0 ( x ) = g 0 ( x ) + h 0 ( x ) Products f ( x ) = g ( x ) × h ( x ) f 0 ( x ) = g 0 ( x ) h ( x ) + g ( x ) h 0 ( x ) Quotient (assume h ( x ) 6 = 0) f ( x ) = g ( x ) h ( x ) f 0 ( x ) = h ( x ) g 0 ( x ) - g ( x ) h 0 ( x ) [ h ( x )] 2 Chain rule: consider the composite function h ( x ) = g f ( x ) = g ( f ( x )) , we can obtain the derivative of h ( x ) with respect to x using the chain rule h ( x ) = g f ( x ) = g ( f (

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