L ectur e 6| 1 Chapter 2
Derivatives (Part 2)
Outline 1. Implicit Differentiation
2. Derivative of Inverse Functions 3. Inverse Trigonometric Functions 4. Linear Approximation
5. Differentials
L ectur e 6| 2 Implicit Differentiation
If a quantity can be expressed in terms of a variable (or quantity) as
we say that is defined explicitly as a function of .
There are, however, many important circumstances where and are related by an equation
such that cannot be solved as a single explicit function of as above.
L ectur e 6| 3 EX Suppose two quantities are related by the equation
If we try to solve in terms of , then
We get two functions and for . Plugging or for , then
is true.
L ectur e 6| 4 EX The equation
gives the so-called the folium of Descartes:
whose graph is as shown below.
The second figure suggests that there are 3 possible functions for that satisfy the equation. However, the formula may not be easy to write down!
L ectur e 6| 5 Def We say is an implicit function induced from if
is true. We also say is implicitly defined from .
We say is an implicit function induced from if
is true. We also say is implicitly defined from .
L ectur e 6| 6 EX and are implicitly defined from the equation
The functions whose graphs are displayed below are implicit functions induced from the equation
Remark For many circumstances, one may not be able to write down formulas of implicit functions!
L ectur e 6| 7 Implicit differentiation
Let be an implicit function induced from the equation
To find we perform the implicit differentiation:
(1) Keep in mind that . (2) Diff w.r.t to get
Don’t forget to use the chain rule.
(3) Solve appeared in (2).
L ectur e 6| 8 EX Find for a function
defined implicitly by
Also calculate at the point .
L ectur e 6| 9 EX Find if
and determine the tangent line at the point .
L ectur e 6| 10 EX Find if
L ectur e 6| 11 Inverse Functions
Given a function , a function is said to be the inverse function of if
is often denoted by .
One can find the inverse function by
L ectur e 6| 12 EX The function
is the inverse function of
because
has no inverse because it is not one-to-one.
?
L ectur e 6| 13 EX (1) Find the inverse function of
(2) Find the inverse function of
L ectur e 6| 14 Derivative of Inverse Functions
Assume is a one-to-one differentiable function with the inverse . If
then is differentiable at and
Thus at any where ,
L ectur e 6| 15 Proof We use implicit differentiation.
Since , this means is implicitly defined by the equation
By implicit diff and the chain rule, we have
So we obtain
hence
L ectur e 6| 16 EX If , find .
L ectur e 6| 17 EX Find the formula for the inverse of the function and find the
derivative .
L ectur e 6| 18 Inverse Trigonometric Functions
with Dom is not 1-1.
It is 1-1 if we consider
Def (The inverse of sine)
Dom and Rng .
L ectur e 6| 19 with Dom is not 1-1.
It is 1-1 if we consider
Def (The inverse of cosine)
Dom and Rng .
L ectur e 6| 20 Derivative Formulas
Proof We show
using implicit diff. That means
L ectur e 6| 21 Def (Other inverse trig functions)
Dom , Rng .
Dom , Rng .
Dom and Rng .
Dom and Rng .
L ectur e 6| 22 Derivative formulas
L ectur e 6| 23 EX (1) Find the value of
(2) Find the limit
Remark The graph of
L ectur e 6| 24 EX Find the derivative
L ectur e 6| 25 Linear Approximation
Let be a function which is differentiable at . Then the following limit exists:
This means
as . Thus when is small, we have the approximation
L ectur e 6| 26
Def The approximation
is called the linear approximation. The function
is called the linearization of at .
Remark For many functions, calculating is difficult whereas
can be easily computed. The error
of this approx. gets smaller as .
L ectur e 6| 27 EX Find the linearization of
at .
Then use it to approximate and .
L ectur e 6| 28 Differentials
For a function , is called an independent variable (i.e. it can take any value freely) and is called a dependent variable (i.e. its value depends on ).
Def (Differentials)
Let be a differentiable function.
The differential of is an independent variable (different from ) denoted by
The differential of is a dependent variable (different from ) denoted by
and it is related to and by
L ectur e 6| 29 Using the independent variable , the linearization at is
By linear approximation,
So is used to approximate the change (or error) in given a change (or error) in :
L ectur e 6| 30 EX Find the differential where
L ectur e 6| 31 EX The circumference of a sphere was
measured to be cm with a possible error of cm.
(1) Use the differential to approximate the maximum error in the calculated area of the sphere.
(2) What is the relative error? What is the percentage error?
Remark
