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DIFFERENSIAL TOTAL Pandang Z = f (x,y) Jika x dan y berubah

DIFFERENSIAL TOTAL Pandang Z = f (x,y) Jika x dan y berubah. x +  x y +  y. x y. Maka z juga berubah :. z +  z = f (x +  x, y +  y) = f (x +  x, y +  y) – z = f (x +  x, y +  y) – f (x,y) z = x² + xy + y². z. z +  z  z Misal.

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DIFFERENSIAL TOTAL Pandang Z = f (x,y) Jika x dan y berubah

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  1. DIFFERENSIAL TOTAL Pandang Z = f (x,y) Jika x dan y berubah x +x y +y x y Maka z juga berubah : z +z = f (x +x, y +y) = f (x +x, y +y) – z = f (x +x, y +y) – f (x,y) z = x² + xy + y² z z +z z Misal z = (x +x)² + (x +x) (y +y) + (y +y)² - x² - xy - y² = … = (2x + y)x + (x + 2y)y Bentuk linier (*) darix dany xy +x² +y² Bentuk non linier (**) dari x dany + Jikax y 0 0 maka (**) 0 Bentuk linier darix dany (*) secara umum : ax + by = a dx + b dy z x z y dengan a= dan b = Page 1 of 8 9 http://www.mercubuana.ac.id kalkulus2_modul-11

  2. c.z = ( 5 – 2 ) ( -0,01 ) + 3 ( 0.02 ) + ( -0,01 ) = - 0,03 + 0,06 – 0,0002 = 0,03 – 0,0002 = 0,0298 dz = ( 5 – 2 ) ( - 0.01 ) + 3 ( 0,02 ) = - 0,03 + 0,06 = 0,03 x = 0,11 y = - 0,05 ~ z + dz d. misalkan x = 3 y=5 z +z = ? z = f ( 3,5 ) = 3.5 – 2.3 = 9 dz = ( y – 2 ) dx + x dy = ( 5 – 2 ) 0,11 + 3 ( - 0,05 ) = 0,18 f ( 3,11 ; 4,95 ) ~ z + dz = 9 + 0,18 = 9,18 Catatan : Jikax dany cukup kecil, maka nilaiz dapat didekati oleh dz Bila z fungsi bervariabel n yaitu z = f (x1, x2, …, xn) maka differensial totalnya : z x n z x1 z x 2 dx2 + … + dz = dx1 + dxn Contoh : W = 2 xy + 5 xz² + y²z + z3 dw = ? Page 3 of 8 9 http://www.mercubuana.ac.id kalkulus2_modul-11

  3. Dalil : Jika z = f (x,y) dan x = g (u,v) y = h (u.v) maka z merupakan fungsi dari u dan v dan z u z v z x z x x u x v z y z y y u y v = = . . + + . . Contoh : x² y² dengan x = uev y = ue-v z = In z u z v Cari dan Secara Umum :  = f (x,y,z, ….) dimana Jika x = f (u, v, w, … ) y = g ( u, v, w, …) z = h (u, v, w, …) ... Maka : x u x v x w y u y v y w z u z v z w  u  v   w  x  x  x  y  y  y  z  z  z +… +… +… = = = . . . + + + . . . + + + . . . Page 5 of 8 9 http://www.mercubuana.ac.id kalkulus2_modul-11

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