4.2 Other techniques
4.2.1 Implicit Differentiation
The equation of a circle
is not in the form
, (although it can be rearranged to
). In this case it is easier to find
directly without rearranging. Differentiate both
sides of the equation
with respect to
, assuming
to be a function of
. We find
Now use
. (Proof: Put
- need
,
.) So
, or
N.B.: This method usually gives
in terms of both
and
.
Example
4.9:
-
Find
for
.
Solution:
-
Differentiating both sides with
respect to
we find
we thus conclude that
4.2.2 Logarithmic
differentiation
If a function has a large number of factors
it may be easier to take the logarithm before
differentiating, using the fat that the logarithm of a
product is the sum of logarithms.
Example
4.10:
-
Find
for
.
Solution:
-
Differentiate both sides with
respect to
:
So
and thus
4.2.3 Differentiation of parametric
equations
Some equations can be written in parametric
form, i.e.,
,
with
a parameter. We can then find its
differential in terms of the parameter. We shall study
this by means of an example only.
Example
4.11:
-
Given circle of radius 4,
use the parametric form to find
and
at
.
Solution:
-
The parametric form is
which clearly satisfies (
4.1
). Now
Note: result is in terms of
. Then
,
(must be in first quadrant), and
therefore
. Now do
.
Note:
Other examples of parametric curves are
- Ellipses
: put
and
,
- Parabola
: put
and
.
- Use of time
, e.g., for
,
.