Math 137 Week 6 Notes – Dr. Paula Smith
3.4
The chain rule gives the derivative of a composition of two functions.
If g is differentiable
at x and f is differentiable at g(x), then f(g(x)) is differentiable at x, and can be evaluated as
dx
dg
dg
df
x
g
x
g
f
x
g
f
=
′
′
=
′
)
(
*
))
(
(
]
))
(
(
[
To get used to this formula, write down f(x), f
′
(x), g(x), g
′
(x), and f
′
(g(x)).
Then the answer is
f
′
(g(x)) * g
′
(x).
If a function is a composition of three functions f, g and h, we can easily extend the chain rule:
dx
dh
dh
dg
dg
df
x
h
x
h
g
x
h
g
f
x
h
g
f
=
′
′
′
=
′
)
(
*
))
(
(
*
)))
(
(
(
]
)))
((
(
(
[
3.5
When y = f(x), we say y is
explicitly
a function of x.
Here, the slope of function y = f(x) is
dx
dy
y
=
′
.
But we can extend this idea to determine the slope of a curve expressed in both x and
y – where y is
implicitly
(one or more) function(s) of x, but y cannot be isolated on one side of
the equation sign.
Given such an equation, take
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 Spring '08
 SPEZIALE
 Math, Calculus, Chain Rule, Derivative, The Chain Rule, dx dx dx, dy dy dy, Dr. Paula Smith, 137 Week

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