Questão resolvida - Derivada de função composta em relação a x, y e z - FAEL

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Tiago Lima - Instagram: @professor_disciplinas_exatas
 
• Seja a função , com , e , ƒ u, v,w = ln|uvw|( ) u = sen xz( ) v = x z2 w = 3x3
determine as derivadas parciais com relação a x, y e z da função.
 
Resolução:
 
 , temos que: = ⋅ ⋅ ⋅
𝜕ƒ u, v, w
x
( ) 1
uvw
𝜕u
𝜕x
𝜕v
𝜕x
𝜕w
𝜕x
= cos xz ⋅ z; = 2xz; = 8x 
𝜕u
𝜕x
( )
𝜕v
𝜕x
𝜕w
𝜕x
2
 
 
Assim, a derivada parcial em relação a x fica: 
 
= ⋅ cos xz ⋅ z ⋅ 2xz ⋅ 8x =
𝜕ƒ u, v, w
x
( ) 1
sen xz ⋅ x z ⋅ 3x( ) 2 3
( ) 2
16cos xz ⋅ z ⋅ x
sen xz z ⋅ 3x
( ) 2 3
( ) 5
 
=
𝜕ƒ u, v, w
x
( ) 16
3
cos xz ⋅ z
sen xz x
( )
( ) 2
 
 , temos que: = ⋅ ⋅ ⋅
𝜕ƒ u, v, w
y
( ) 1
uvw
𝜕u
𝜕y
𝜕v
𝜕y
𝜕w
𝜕y
= 0; = 0; = 0 
𝜕u
𝜕y
𝜕v
𝜕y
𝜕w
𝜕y
 
 
Assim, a derivada parcial em relação a y fica: 
 
= ⋅ 0 ⋅ 0 ⋅ 0
𝜕ƒ u, v, w
y
( ) 1
sen xz ⋅ x z ⋅ 3x( ) 2 3
 
= 0
𝜕ƒ u, v, w
y
( )
 
 , temos que: = ⋅ ⋅ ⋅
𝜕ƒ u, v, w
z
( ) 1
uvw
𝜕u
𝜕z
𝜕v
𝜕z
𝜕w
𝜕z
= cos xz ⋅ x; = x ; = 0 
𝜕u
𝜕z
( )
𝜕v
𝜕z
2
𝜕w
𝜕z
 
Assim, a derivada parcial em relação a z fica: 
 
= ⋅ cos xz ⋅ x ⋅ x ⋅ 0
𝜕ƒ u, v, w
z
( ) 1
sen xz ⋅ x z ⋅ 3x( ) 2 3
( ) 2
 
 
 
= 0
𝜕ƒ u, v, w
z
( )