Transcript Section 13.5 Chain Rules for Functions of Several Variables

Section 13.5
Chain Rules for Functions of Several
Variables
Theorem 13.6 Chain Rule:
One Independent Variable
and Figure 13.39
Examples
Find dw using the appropriate Chain Rule.
dt
y
1) w  ln
x
w  xyz
2)
xt
2
y  2t
ze
t
x  cos t
y  sin t
Example
Differentiate implicitly to find the first partial derivatives
of w.
z  e sin( x  y)
x
Theorem 13.7 Chain Rule:
Two Independent
Variables and Figure 13.41
Example:
w
Find s and
Rule
w=xcosyz
w
tusing the appropriate Chain
x  s2
y t
2
z  st
2
Theorem 13.8 Chain Rule:
Implicit Differentiation