Tiago Lima - Instagram: @professor_disciplinas_exatas
• Seja , em que , e , determine .
f
(
x
,
y
,
z
) = 2
x
+
y
+ 5
z
3 4 2
x
=
t
y
=
t
²
z
=
t
³
df
dt
Resolução:
Devemos aplicar a regra da cadeia;
=
⋅
+
⋅
+
⋅
df
dt
𝜕
f
𝜕
x
𝜕
x
𝜕
t
𝜕
f
𝜕
y
𝜕
y
𝜕
t
𝜕
f
𝜕
z
𝜕
z
𝜕
t
= 6
x
𝜕
f
𝜕
x
2
= 1
𝜕
x
𝜕
t
= 4
y
𝜕
f
𝜕
y
3
= 2
t
𝜕
y
𝜕
t
= 10
z
𝜕
f
𝜕
z
2
= 3
t
𝜕
z
𝜕
t
2
Assim, a derivada fica:
= 6
x
⋅
1 + 4
y
⋅
2
t
+ 10
z
⋅
3
t
df
dt
2 3 2 2
como
:
x
=
t
,
y
=
t
²
e
z
=
t
³
= 6
t
⋅
1 + 4
t
²
⋅
2
t
+ 10
t
³
⋅
3
t
= 6
t
+ 4
t
⋅
2
t
+ 10
t
⋅
3
t
df
dt
2( )3( )2 2 2 6 6 2
= 6
t
+ 8
t
+ 30
t
=
30t
+
8t
+
6t
df
dt
2 7 8
872
