Differentiation Rules (2A) Young Won Lim 1/30/16

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 ) = ( f x g x g f g )( 1 ) x g 2 Differentiation Rules (2A) 4

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

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

7 Derivative Prouct an Quotient Rule f g x f ' g + f g' f (x), g(x) x 2 x 3 (2 x) x 3 + x 2 (3 x 2 ) x e 2 x e 3 x (2 e 2 x ) e 3 x + e 2 x (3 e 3 x ) x Differentiation Rules (2A) 7

8 Chain Rule Differentiation Rules (2A) 8

9 Chain Rule f (g(x)) x f ' (g(x)) g' (x) f x = f g g x f g = f ' (g( x)) g x = g ' (x) f (g(x)) x f ' (g(x)) g' (x) with respect to with respect to x e P(x )x x e P(x )x x ( P(x)x ) = e P(x)x P(x) e g x e g g x f g g x Differentiation Rules (2A) 9

10 Chain Rule an Substitution Rule Examples Differentiation Rules (2A) 10

11 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) 11

12 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) 12

13 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) 13

14 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) 14

15 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) 15

16 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) 16

17 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