Week 11d - QMA 2006S2 (Differentiation & Marginal Concepts)

Week 11d - QMA 2006S2 (Differentiation & Marginal Concepts)

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Unformatted text preview: Jessica Chung & Derek Hui Week 11 QMA PASS | 2006 S2 | Tues, 3-4pm | QUAD G042 Questions? Email us: [email protected] , [email protected] 1/4 REVISION TOPICS Limits, continuity and derivative rules should all be familiar to you! Make sure you don’t forget to revise them though... DERIVATIVES OF LOGARITHMIC FUNCTIONS IF: y = ln x y = ln u (where u is a positive differentiable function of x) y = log b u (where u is a positive differentiable function of x) (ln ) 1 d x dx x = (ln ) 1 d u du dx u dx = × ( ) ln (ln ) ln 1 1 1 ln ln u d d u b du dx b dx b u dx = × = × × DERIVATIVES OF EXPONENTIAL FUNCTIONS IF: y = e x y = e u ( ) x x d e e dx = ( ) u u d e du e dx dx = × y = a x y = a u ( ) ln x x d a a a dx = × ( ) ln u u d a du a a dx dx = × × EXAMPLES: Find the derivative if 1. y = 2 ln x 1 2 2 dy dx x x = × = 2. y = 5e 2x+1 2 1 2 1 5 2 10 x x dy e dx e + + = × × = 3. y = e x lnx ( ) ( ) ( ) ln ln 1 ln 1 1 ln x x x x dy x x e dx x x e & ¡ = × + × ¢ £ ¤ ¥ = + 4. y = log 5 (x 4- x + 6ln2x) ( ) 3 4 3 4 1 1 1 4 1 6 ln 5 6ln 2 6 4 1 1 ln 5 6ln 2 dy x dx x x x x x x x x x & ¡ = × ×- + × ¢ £ ¤ ¥- +- + = ×- + IMPLICIT DIFFERENTIATION Implicit differentiation is used to differentiate functions involving both x and y but where it is difficult to make x a function of y or vice versa. Rules for implicit differentiation - Differentiate x terms as usual - Differentiate y terms the same way, and multiply by dy dx- Use the product rule for terms involving both x and y EXAMPLE: Find dy dx if x 2 + y 2- 2xy = 0 ( ) 2 2 2 1 (2) 2 2 2 2 (2 2 ) 2 2 2 2 2 2 1 dy dy x y x y dx dx dy dy x y x y dx dx dy y x y x dx dy y x dx y x & ¡ +- ⋅ + = ¢ £ ¤ ¥ +-- =- =-- =- = DIFFERENTIALS These points are just for your understanding of differentials & the next section - we’ve never come across an exam explicitly testing these concepts!...
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This note was uploaded on 04/02/2012 for the course ECON 1101 taught by Professor Julia during the Three '08 term at University of New South Wales.

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Week 11d - QMA 2006S2 (Differentiation & Marginal Concepts)

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