1. Frank Cowell: Differentiation DIFFERENTIATION MICROECONOMICS Principles and Analysis Frank Cowell July 2015 1

2. Frank Cowell: Differentiation Overview... July 2015 2 Basics Chain rule Elasticities Differentiation Basic definitions l’Hôpital’s rule

3. Frank Cowell: Differentiation Definition (1) July 2015 3

4. Frank Cowell: Differentiation Examples  Take an example where the limit is easy to evaluate  Let f(x) = x 2  So f(x +  x) = x 2 + 2x  x + [  x] 2  Therefore [f(x +  x)  f(x)] /  x = 2x +  x  Take the limit of this as  x  0 clearly we have df(x)/dx = 2x  Some other examples July 2015 4

5. Frank Cowell: Differentiation Definition (2) July 2015 5

6. Frank Cowell: Differentiation Overview... July 2015 6 Basics Chain rule Elasticities Differentiation Differentiation involving a “function of a function” l’Hôpital’s rule

7. Frank Cowell: Differentiation Chain rule (1) July 2015 7

8. Frank Cowell: Differentiation Chain rule (2)  For  x ≠ 0 and  y ≠ 0 we could write  z/  x = (  z/  y)  (  y/  x) follows from simple rearrangement of expressions on previous page  Now assume that f and  are differentiable as  x  0, we have  y  0 and… [f(x+  x)  f(x)]/  x becomes f '(x), [g(x+  x)  g(x)]/  x becomes g'(x) as  y  0, [  (y+  y)   (y)]/  y becomes  '(y)  Drawing these results together in the limit we have the differential of g(x) =  (f(x)): g'(x) =  '(y)  f '(x)  This chain rule can be extended indefinitely to the function of a function of a function of a… July 2015 8

9. Frank Cowell: Differentiation Chain rule (3) July 2015 9

10. Frank Cowell: Differentiation Basics Overview... July 2015 10 Chain rule Elasticities Differentiation Practical application l’Hôpital’s rule

11. Frank Cowell: Differentiation Elasticity (1) July 2015 11

12. Frank Cowell: Differentiation Elasticity (2)  Consider a natural multivariate extension  Suppose we have differentiable f 1, …,f m such that  Then the elasticity of y j with respect to x i is  An equivalent way of writing this elasticity is July 2015 12

13. Frank Cowell: Differentiation Basics Overview... July 2015 13 Chain rule Elasticities Differentiation A useful result l’Hôpital’s rule

14. Frank Cowell: Differentiation l’Hôpital’s rule July 2015 14