Calc_Review - ECON 100A Intermediate Microeconomics...

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ECON 100A: Intermediate Microeconomics CALCULUS REVIEW In this course we will encounter problems requiring calculus, especially dif- ferentiation. I have prepared the following as a review of the material you have learnt in a basic course on calculus. 1. Simple Power Functions y = x a dy dx = ax a ° 1 For example if a = 1, then dy dx = 1 ° x 0 = 1 : If a = 10, then dy dx = 10 ° x 9 : 2. Sums, Di/erences and Constants: Example 1: y = x 2 + 2 x 3 + 5 dy dx = 2 ° x + 6 ° x 2 Example 2: y = x 2 ± 2 x 3 ± 5 dy dx = 2 ° x ± 6 ° x 2 Note: The derivative of a constant is zero. 3. Product Rule y ( x ) = y 1 ( x ) ° y 2 ( x ) dy dx = y 1 ( x ) ° dy 2 dx + y 2 ( x ) ° dy 1 dx Note: The hi-ho rule can be used as an aid. The derivative of hi-ho equals, hi dee ho plus ho dee hi. Here, hi is the function y 1 ( x ) ; ho is the function y 2 ( x ) and dee refers to taking the derivative. 1
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Example: y = x 4 x 5 dy dx = 4 x 3 ° x 5 + 5 x 4 ° x 4 dy dx = 9 x 8 Similarly, using the simple power function example where y = x 9 , we have: dy dx = 9 x 8 We will be seeing a lot of functions where there maybe two di/erent variables, x 1 and x 2 ; for example y = x 4 1 x 5 2 : Here, we will not be able to use the simple power function rule as adding x 1 and x 2 is like adding apples and oranges.
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