Tuesday, October 4
−
Lecture 12 :
Implicit differentiation, logarithmic differentiation,
higherorder derivatives
(Refers to section 3.8 and 4.3 pp. 26064)
Students who have mastered the content of this lecture know about
:
The derivatives of log functions
,
logarithmic differentiation, higherorder derivatives
Students who have practiced the techniques presented in this lecture will be able to
:
Perform
implicit differentiation of one variable with respect to another when this variable is implicitly defined
in an equation
,
determine the derivative of a log function
,
differentiate a function by logarithmic
differentiation
,
recognize those functions which are best differentiated by logarithmic differentiation
.
12.1
Introduction
We have studied the derivative of functions
y
=
f
(
x
) where
y
is
expressed explicitly as a function of
x
. However there are functions where
y
is implicitly
defined as function of
x
. For example,
x
2
+
y
2
= 1. From this equation we in fact extract
from it two functions where
y
is a function of
x
:
In spite of this we may still want to find the equation of the tangent to the curve (in this
case a unit circle) at the point say, (
√
2/2,
√
2/2).
We would first want to determine to
which of the above two functions this point belong to, and then differentiate that function.
However we can proceed more quickly by directly differentiating the equation
x
2
+
y
2
= 1
with respect to
x
as follows:
12.1.1
Definition
−
We refer to the method of differentiation described above example
as
implicit differentiation
.
It is often the only way we can determine the slope of a tangent line to a curve at a
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 Spring '11
 RobertAndre
 Calculus, Derivative, Implicit Differentiation, Logarithm

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