Differentiation Rules (2A) Young Won Lim 2/22/16

Transcription

1 Differentiation Rules (2A)

2 Copyright (c) Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any later version publishe by the Free Software Founation; with no Invariant Sections, no Front-Cover Texts, an no Back-Cover Texts. A copy of the license is inclue in the section entitle "GNU Free Documentation License". Please sen corrections (or suggestions) to youngwlim@hotmail.com. This ocument was prouce by using LibreOffice an Octave.

3 Derivative Prouct an Quotient Rule Differentiation Rules (2A) 3

4 Prouct an Quotient Rule f g x f ' g + f g' f (x), g(x) x (f g) = ( f x g + f g ) x f g x f ' g f g' g 2 f (x), g(x) ( f ) x g = ( f x g f g )( 1 2) x g Differentiation Rules (2A) 4

5 Prouct Rule f g x f ' g + f g' f (x), g(x) f (x)g(x) x f '(x)g(x) + f (x)g' (x) x (f g) = f g g + f x x (f g) = f g + f g x (f g )x = ( f x g ) x + ( f g x ) x = ( f x x ) g + f ( g x x ) = f g + f g Differentiation Rules (2A) 5

6 Quotient Rule f (x)g( x) x f ' (x) g( x) + f ( x) g' (x) 1 f (x){ } 1 f ' g( x) (x){ } g( x) + f ( x) { g(x)} 1 x ' 1 = f ' (x){ } g( x) + f ( x) { g' (x) } g 2 (x) f ' ( x) g(x) f ( x) g '( x) = g 2 ( x) x g 1 (x) = g 2 (x) x g(x) Differentiation Rules (2A) 6

7 Example 1 f g x f ' g + f g' x 2 x 3 (2 x) x 3 + x 2 (3 x 2 ) x x x5 = 5 x 4 e 2 x e 3 x (2e 2 x ) e 3 x + e 2x (3 e 3 x ) x x e5 x = 5e 5 x Differentiation Rules (2A) 7

8 Example 2 x 2 cos(x) x (2 x) cos(x) x 2 ( sin(x)) cos(x) = 1 1 2! x ! x4 1 6! x6 + sin (x) = x 1 3! x ! x5 1 7! x7 + x 2 cos(x) = x 2 (1 1 2! x ! x4 1 6! x6 + ) x x2 cos(x) = 2 x (1 1 2! x ! x4 1 6! x6 + ) + x 2 ( x + 1 3! x3 1 5! x ! x7 ) Differentiation Rules (2A) 8

9 Example 3 (x+ p)(x+q) (x+q) + (x + p) x x 2 +(p+q)x+ pq 2 x+(p+q) 1 (x+ p)(x+q) x (x+q) + (x+ p) (x+ p) 2 (x+q) 2 (x+r) (x+ p)(x+q) 1 (x+r) (x+ p)(x+q) x (x+ p)(x+q) (x+r)((x+q) + (x+ p)) 1 (x+ p)(x+q) (x+ p) 2 (x+q) 2 (x+r) (x+q) + (x+ p) (x+ p) 2 (x+q) 2 Differentiation Rules (2A) 9

10 x x n+1 = x (xn x) = ( x x n ) x + xn ( x x ) = (n x n 1 ) x + x n 1 = (n+1)x n x xn+1 = (n+1)x n+1 1 x xn = n x n 1 Differentiation Rules (2A) 10

11 Differentiation Rules (2A) 11

12 Differentiation Rules (2A) 12

13 Differentiation Rules (2A) 13

14 f (x) = 1 x g( x ) = sin( x) f (x ) g( x) = sin(x) x h( x) = h1( x) + h2( x) h1( x) = x 1 x sin( x) = 1 sin( x) 2 x h2(x) = 1 x x sin( x) = 1 cos( x) x Differentiation Rules (2A) 14

15 Differentiation Rules (2A) 15

16 Differentiation Rules (2A) 16

17 Differentiation Rules (2A) 17

18 Differentiation Rules (2A) 18

19 Differentiation Rules (2A) 19

20 Chain Rule Differentiation Rules (2A) 20

21 Chain Rule f (g(x)) x f ' (g) g' (x) f (g) = f (g( x)) f x = f g g x f (g(x)) x f ' (g(x)) g' (x) with respect to with respect to x Differentiation Rules (2A) 21

22 Chain Rule x y = f (x) y f (x) x y=g(x) y g(x) x u=g(x) u y= f (u) f (u) f (g(x)) g x f g f x = f g g x Differentiation Rules (2A) 22

23 Example 1 u=g(x) y= f (u) x x 3 a( x 3 ) 2 +b = a x 6 +b g(x) = x 3 f (x) = a x 2 +b g ' (x) = 3 x 2 f ' (g) = 2a g f ' (g) g' (x) = (2 a g) (3 x 2 ) = (2a x 3 ) (3 x 2 ) = 6 a x 5 Differentiation Rules (2A) 23

24 Example 2 u=g(x) y= f (u) x f (g(x)) = f ' (x)g'(x) x g(x) f (x) = a x 2 +b f ' (x) = 2a x g(x) = x 3 g ' (x) = 3 x 2 f ' (g(x))g ' (x) 2a(x 3 ) 3 x 2 Differentiation Rules (2A) 24

25 Chain Rule an Substitution Rule Examples Differentiation Rules (2A) 25

26 Chain Rule an Substitution Rule f (g(x)) f x x = f ' (g(x)) g' (x) f g g x f (g(x)) + C x f ' (g(x)) g' (x) f (g(x)) + C = f ' (g(x)) g'(x)x Differentiation Rules (2A) 26

27 Chain Rule an Substitution Rule Examples e ( x2 +2) f x = f g g x = eg (2 x) = e x 2 +2 (2 x) f (x)=e x g(x)=x 2 +2 f (g)=e g f g =eg g x =2 x e x2 +2 (2 x)x f g g=eg = e x c or f g g=eg = e x c view (I) f '(x)=e x g(x)=x 2 +2 f '(g)=e g f g g g x=2x x x f '(g)=e g +c view (II) f (x)=e x g(x)=x 2 +2 f (g)=e g f (g)g g x x=2x x F (g)=e g +c Differentiation Rules (2A) 27

28 Chain Rule x y = f (x) y x y=g(x) y x u=g(x) u y = f (u) y y x = y u u x Differentiation Rules (2A) 28

29 Chain Rule an Partial Differentiation z= f (x, y) (x, y) y t x=g(t) x t y=h(t) y y =h(t) t x=g(t) (x, y) z= f (x, y) y z t = z x x t + z y y t Differentiation Rules (2A) 29

30 Chain Rule an Total Differentials z = f (x, y) x(t) y (t) z = z x x + z y y x = x t t y = y t t z t t = z x x t t + z y y t t z t = z x x t + z y y t Differentiation Rules (2A) 30

31 Parameterize Function of Two Variables vative f (x, y ) (x, y) z f (x, y) x = f x = z x (1,1,3) f (x, y) y = f y = z y z = x 2 + x y + y 2 z x = 2x + y z = x 2 + x y + y 2 z y = x + 2 y t x(t) (x, y ) f (x, y ) z y (t) w(t) z = f x x + f y y t z w t (t) = z t = f x x t + f y y t Differentiation Rules (2A) 31

32 References [1] [2] M.L. Boas, Mathematical Methos in the Physical Sciences [3] E. Kreyszig, Avance Engineering Mathematics [4] D. G. Zill, W. S. Wright, Avance Engineering Mathematics