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Unformatted text preview: QABE Lecture 15 Differentiation II: Tricks and Extensions School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Implicit Differentiation 2 2.1 Implicit functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Differentiation of implicit functions . . . . . . . . . . . . . . . . . . . . . . 3 3 Logs and Exponentials 4 4 Higher Order Derivatives 6 5 Elasticity of Demand 7 1 Introduction We continue our introduction to the nature of change . Last time, we looked at a variety of functions and how we might compute their derivative an expression for the rate of change of the dependent variable in terms of the independent variable at a given point. Recall that we arrived at these expressions by applying our understandings of limits to the problem. We took a secant on the function we were dealing with, and drew this closer and closer to the point of interest, until it was so close (the distance between our point of interest and the intersection with the function at some other point) that we could see that the line we were drawing actually made a very good approximation to the tangent to the function at that point just what we were after. This time, we will need this apparatus to look at three possibly tricky problems in differentiation. The first, differentiation of implicit functions, forces us to consider at a deeper level what we are actually doing with our small dee-y, dee-x fraction symbol. In particular, well see that we can treat it like another variable in our equation, and therefore, solve for it! In the second problem, we consider the slightly different logarithmic and exponential functions in terms of their derivatives (and actually find them to be highly related). Finally, we touch on the seemingly dicult, but actually quite simple area of multiple derivatives that is, doing our differentiation work more than once on the same func- tion. For now, well just see the mechanics of the process, but in the next lecture, well need this to help us identify the best or worst point on a function.need this to help us identify the best or worst point on a function....
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