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IMPLICIT DIFFERENTIATION
Equations in terms of
x
and
y
can be defined two different ways. One is explicitly,
y
=
f
(
x
), and the
other, implicitly.
EXPLICITLY:
y
=
x
2
+ 3
x
 4
IMPLICITLY:
x
2
+
y
2
= 4
We know how to take the derivative of the first.
y
=
x
2
3
x
−
4
⇒
y '
=
2
x
3
If we want to find the derivative of the implicitly defined function, we would first have to solve for
y
in terms of
x
.
Then the question would be this: which function would we want to use? It would
depend on what we were asked to find. As you should know, calculus is not a guessing game, so there
has to be an easier way to find the derivative. The easy way to find this derivative is to find it by
implicit differentiation.
When performing implicit differentiation, you must remember that
y
is a function of
x
. Hence,
every time you take the derivative of
y
, you are performing a chain rule. Therefore, you must tack on a
dy
dx
(or a
y
prime). After taking the derivatives, solve for
dy
dx
in terms of
x
and
y
.
EXAMPLE 1:
SOLUTION:
EXAMPLE 2:
Find
dy
dx
for
3
x y
7
2
= 6
y
.
SOLUTION:
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 Spring '08
 Noohi
 Equations, Derivative, Implicit Differentiation

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