Differentiation Rules (2A)
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Derivative Prouct an Quotient Rule Differentiation Rules (2A) 3
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
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
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
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
Example 2 x 2 cos(x) x (2 x) cos(x) x 2 ( sin(x)) cos(x) = 1 1 2! x2 + 1 4! x4 1 6! x6 + sin (x) = x 1 3! x3 + 1 5! x5 1 7! x7 + x 2 cos(x) = x 2 (1 1 2! x2 + 1 4! x4 1 6! x6 + ) x x2 cos(x) = 2 x (1 1 2! x2 + 1 4! x4 1 6! x6 + ) + x 2 ( x + 1 3! x3 1 5! x5 + 1 7! x7 ) Differentiation Rules (2A) 8
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
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
Differentiation Rules (2A) 11
Differentiation Rules (2A) 12
Differentiation Rules (2A) 13
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
Differentiation Rules (2A) 15
Differentiation Rules (2A) 16
Differentiation Rules (2A) 17
Differentiation Rules (2A) 18
Differentiation Rules (2A) 19
Chain Rule Differentiation Rules (2A) 20
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
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
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
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
Chain Rule an Substitution Rule Examples Differentiation Rules (2A) 25
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
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 2 +2 + c or f g g=eg = e x 2 +2 + 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
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
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
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
Parameterize Function of Two Variables http://en.wikipeia.org/wiki/partial_eri 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
References [1] http://en.wikipeia.org/ [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