Week 8
The Chain Rule
The Chain Rule for
z
=
f
(
x
(
t
)
, y
(
t
))
In two variables, there are different kinds of Chain Rules; the one we will see is for situations
like
z
=
f
(
x
(
t
)
, y
(
t
)), where
z
is a function of
x
and
y
, and
x
and
y
are both functions of
t
. In such a situation,
t
is called the
independent variable
, because it doesn’t depend on any
other variable, while
x
,
y
and
z
are called
dependent variables
, since they depend on other
variables.
Recall that one way to write the 1variable Chain Rule for
y
=
f
(
x
(
t
)) is
dy
dt
=
dy
dx
dx
dt
. The
same notation works best for the 2variable Chain Rule with 1 independent variable:
dz
dt
=
∂z
∂x
dx
dt
+
∂z
∂y
dy
dt
Note that since
z
=
f
(
x, y
) is a 2variable function,
∂z
∂x
and
∂z
∂y
are partial derivatives, so are
written with
∂
’s, while the other derivatives are of 1variable functions, so are written with
d
’s.
This notation is nice and short, but can be a bit misleading; we should also write it out more
explicitly, with
z
=
g
(
t
) denoting
z
directly as a function of
t
:
z
=
g
(
t
) =
f
(
x
(
t
)
, y
(
t
))
⇒
dz
dt
=
g
0
(
t
) =
f
x
(
x
(
t
)
, y
(
t
))
·
x
0
(
t
) +
f
y
(
x
(
t
)
, y
(
t
))
·
y
0
(
t
)
.
Then if we plug in a value
t
=
a
, we get:
dz
dt
t
=
a
=
g
0
(
a
) =
f
x
(
x
(
a
)
, y
(
a
))
·
x
0
(
a
) +
f
y
(
x
(
a
)
, y
(
a
))
·
y
0
(
a
)
.
Note that in theory we could always write out
z
=
g
(
t
) explicitly, and differentiate that as a
1variable function; but in practice this might be too much work, since this will be a more
complicated function, hence more work to differentiate.
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 Fall '10
 MalabikaPramanik
 Calculus, Chain Rule, Derivative, The Chain Rule, dt

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