Derivative
Advanced Level Pure Mathematics
Calculus I
Chapter 4 Derivatives
4.1 Introduction 2
4.2 Differentiability 4
Continuity and Differentiability 6
4.3 Rules of Differentiation 9
4.5 Higher Derivatives 13
4.6 Mean Value Theorem 17
4.1 Introduction
Let [pic] be a fixed point and [pic] be a variable point on the curve [pic] as shown on about figure. Then the slope of the line AP is given by [pic] or [pic]. When the variable point P moves closer and closer to A along the curve [pic], i.e. [pic]. the line AP becomes the tangent line of the curve at the point A. Hence, the slope of the tangent line at the point A is equal to [pic]. This term is defined to be the derivative of [pic] at [pic] and is usually denoted by [pic]. The definition of derivative at any point x may be defined as follows.
Definition Let [pic] be a function defined on the interval [pic] and [pic].
[pic] is said to be differentiable at [pic] ( or have a derivative at [pic] ) if the limit [pic] exists. This lime value is denoted by [pic] or [pic] and is called the derivative of [pic] at [pic].
If [pic] has a derivative at every point x in [pic], then [pic] is said to be differentiable on [pic].
Remark As [pic], the difference between x and [pic] is very small, i.e. [pic] tends to zero. Usually, this difference is denoted by h or [pic]. Then the derivative at [pic] may be rewritten as [pic]. ( First Principle )
Example Let [pic]. Find [pic].
Example Let [pic]. Find [pic].
Example If [pic], find [pic] .
Example Let f be a real-valued function defined on R such that for all real numbers x and y, [pic]. Suppose f is differentiable at [pic], where [pic].
(a) Find the value of [pic].
(b) Show that f is differentiable on R and express [pic] in terms of [pic].
4.2 Differentiability
Example Let [pic]. Show that [pic], f is continuous but not differentiable.
Solution
By definition, [pic] = [pic] = [pic]
= [pic]
= [pic].
Since [pic] does not exist, [pic] is not differentiable at [pic].
Example If [pic], show that [pic], f is continuous but not differentiable.
Example Let [pic].
Find [pic] and [pic] by definition. Does [pic] exist? Why?
Example Show that [pic] is not differentiable at [pic]. Find also the derivative of [pic] when [pic].
Example Let [pic] Find [pic].
Example Let [pic]. Find [pic].
Example A function [pic] is defined as [pic].
Find a, b ( in term of c ) if [pic] exists.
Continuity and Differentiability
Theorem Let [pic]. If [pic] exist, then [pic] implies that [pic].
Proof Since [pic], we have[pic] [pic]
[pic] ( Since both limit exist )
[pic].
As [pic] exists, [pic] is bounded. Futhermore, [pic] and so [pic].
Theorem If [pic] is differentiable at [pic] then [pic] is continuous at [pic] .
Proof By Theorem, [pic], i.e.[pic].
Hence, [pic] is continuous at [pic].
Remark We should have a clear concept about the difference between
(a) [pic] is well-defined at [pic].
(b) the limit of [pic] at [pic] exists.
(c) [pic] is continuous at [pic].
(d) [pic] is differentiable at [pic].
D [pic] C [pic] L
Example Let [pic].
Show [pic] is continuous at [pic] and discuss the continuity of [pic] at [pic].
Example Prove that if [pic] satisfy [pic] [pic] and [pic] where [pic], then [pic] exist [pic] and [pic]. Find [pic]
Example Prove that if [pic] satisfies [pic], then [pic], where [pic] and
find [pic].
Example Find a, b in terms of c for [pic] exists where [pic]
Example Let [pic] be a real-valued function such that
[pic] [pic]
Show that [pic] for all [pic].
4.3 Rules of Differentiation
Composite functions
|[pic] |[pic] |[pic] |
| | | |
|[pic] | | |
| |[pic] |[pic] |
Algebraic functions
[pic] where k must be independent of x (usually a constant)
Inverse functions (esp.: inverse of trigo func)
[pic]
Trigonometric functions
|[pic] |[pic] |[pic] |
| | | |
|[pic] |[pic] |[pic] |
Logarithmic functions
|[pic] |[pic] |
| | |
|[pic] |[pic] |
Parametric functions (commonly use in Rate of change)
[pic]
Theorem Chain Rule
If [pic] , i.e. [pic] and f, g are differentiable, then [pic].
Example [pic], [pic] =
=
= [pic]
Example Find the derivatives of the following functions:
(a) [pic] (b) [pic]
(c)[pic] (d) [pic], where [pic]
Example ( Derivatives of inverse )
Let [pic] find [pic].
Example ( Derivatives of inverse function )
Prove [pic]
Solution Let [pic]. [pic] = [pic]
[pic] = [pic]
[pic] = [pic]
[pic] [pic][pic] = [pic]
Example Prove [pic] = [pic]
Solution
Remark [pic] = [pic] , [pic] = [pic]
[pic] = [pic] , [pic] = [pic]
[pic] = [pic][pic] , [pic] = [pic]
Example* (a) Find [pic] (b) Find [pic]
Example Find [pic] if
(a) [pic], where a is a constant.
(b) [pic].
Example Find the derivative of following functions
(a) [pic] (b) [pic]
Example If [pic] and [pic], find [pic]
Exercise 4B 4(c), (d), (i), (k), (n), (q), (n), (v), (w), (x), (y), (z)
4.5 Higher Derivatives
Definition If [pic] is a function of [pic], then the nth derivatives of y w.r.t. x is defined as [pic] if [pic] is differentiable.
Symbolically, the nth derivatives of y w.r.t. x is denoted by [pic] or [pic].
Remark 1. [pic][pic] but [pic]
[pic]
2. If [pic] is function of [pic], [pic], then [pic] (.[pic]).
[pic]=[pic] = [pic]
= [pic]
= [pic]
Example Let [pic]. Find [pic].
Example Prove that [pic] satisfies the equation [pic].
Example If [pic], where u is a function of x, Prove [pic] where [pic] and [pic] are the nth derivative of y and u respectively.
Hence, if [pic], prove [pic].
Example Find a general formula for the [pic]th derivative of
(a) [pic] ([pic])
(b) [pic]
(c) [pic] [pic]
Theorem Let [pic] and [pic] be two functions which are both differentiable up to nth order. Then
(a) [pic]
(b) [pic]
Theorem Leibniz's Theorem
Let [pic] and [pic] be two functions with nth derivative. Then
[pic] where [pic].
Example Let [pic], where a is a real constant. Find [pic].
Example Find [pic], [pic].
Example Let [pic]. Show that for [pic], [pic].
Example Prove that if [pic] then [pic]
Deduce that when [pic] [pic]
Example (a) Prove if [pic] , [pic].
(b) Show if [pic], then [pic].
4.6 Mean Value Theorem
Definition Let [pic] be a function defined on an interval I. f is said to have an absolute maximum at c if [pic]I and [pic] is called the absolute maximum value.
Similarly, f is said to have an absolute minimum at d if [pic]I and [pic] is called the absolute minimum value.
Theorem Fermat's Theorem
Let [pic] be defined and differentiable on an open interval (a, b). If [pic] attains its absolute maximum or absolute minimum (both are called absolute extremum) at [pic], where [pic], then [pic].
Proof For any [pic] there exists a real number h such that [pic] and [pic]. Now, suppose [pic] attains its absolute maximum at [pic]. Then we have [pic] and [pic] , and so [pic] and [pic]. Now, the left and right hand derivatives are given by
[pic], ( since [pic] )
and [pic]. ( since [pic] )
Since [pic] is differentiable at [pic], the left and right hand derivatives must be equal,
i.e. [pic]. This is possible only if [pic].
The proof for [pic] attaining its absolute minimum at [pic] is similar and is left as an exercise.
Remark 1. [pic] NOT IMPLIES absolute max. or min. at [pic].
e.g. [pic] at [pic], not max. and min.
figure
2. Fermat's Theorem can't apply to function in closed interval.[pic]absolute max. or min may be attained at the end-points. As a result, one of the left and right hand derivatives at c may not exist.
e.g. [pic]defined on [ 0, 5] attains its absolute max. at [pic] but its right hand derivative does not exist.
3. Fermat' s Theorem can't apply to function which are not differentiable.
e.g. [pic] . Not differentiable at [pic] but min. at [pic].
figure
Theorem Rolle's Theorem
If a function [pic] satisfies all the following three conditions:
(1) [pic] is continuous on the closed interval [pic],
(2) [pic] is differentiable in the open interval [pic],
(3) [pic];
then there exists at least a point [pic] such that [pic].
Proof Since [pic] is continuous on [pic] [pic] [pic] is bounded
(i) [pic], where m (min), M (Max) are constant.
[pic] [pic]
[pic] [pic]
[pic] [pic]
(ii) [pic], the max. and min. cannot both occur at the end points a, b.
[pic] [pic] such that [pic]
i.e. [pic] [pic] sufficiently closed to p.
By Fermat's Theorem, [pic] exist and equal to 0.
Example Define [pic] on [0,4]. Note that [pic].
We have [pic] and so [pic]. Since [pic], Rolle's Theorem is verified.
The geometric significance of Rolle's theorem is illustrated in the following diagram.
If the line joining the end points [pic] and [pic] is horizontal (i.e. parallel to the x-axis) then there must be at least a point [pic] (or more than one point) lying between a and b such that the tangent at this point is horizontal.
Theorem Mean Value Theorem
If a function [pic] is
(1) continuous on the closed interval [pic] and
(2) differentiable in the open interval [pic],
then there exists at least a point [pic] such that
[pic].
Proof Consider the function g defined by
[pic] [pic] [pic] is differentiable and continuous on [pic].
Let [pic]
[pic] [pic] is also differentiable and continuous on [pic].
We have [pic]
[pic] By Rolle's Theorem, [pic] such that [pic]
[pic] [pic]
[pic] [pic]
Remark: 1. The Mean Value Theorem still holds for [pic]. [pic].
2. Another form of Mean Value Theorem [pic]
3. The value of p can be expressed as [pic] , [pic].
[pic] [pic]
Example Use the Mean Value Theorem. show [pic]
(a) [pic]
(b) [pic], [pic]
(c) [pic], [pic].
Example By using Mean Value Theorem, show that
[pic]
for all real values [pic] and [pic].
Solution Let [pic].
Case (i)
Case (ii)
Case (iii)
Example Let [pic] such that [pic] and [pic] be a differentiable function on [pic] such that [pic], [pic] and [pic] is strictly decreasing. Show that [pic].
Example Let [pic] be a continuous function defined on [ 3, 6 ]. If [pic] is differentiable on ( 3, 6 ) and [pic]. Show that [pic].
Example Let [pic] be a polynomial with real coefficients.
If [pic] by using Mean Value Theorem, show that the equation [pic] has at least one real root between 0 and 1.
Example Let [pic] be a real-valued function defined on [pic]. If [pic] is an increasing function,
show that [pic] [pic]
Example Let [pic] be a real-valued function such that
[pic], [pic]
Show that [pic] is a differentiable function.
Hence deduce that [pic] for all [pic], where [pic] is a real constant.
Example Let [pic] be a function such that [pic] is strictly increasing for [pic].
(a) Using Mean Value Theorem, or otherwise, show that
[pic]
(b) Hence, deduce that
[pic]
Theorem Generalized Mean Value Theorem
Let [pic] and [pic] such that
(i) [pic] and [pic] are continuous on [ a, b ].
(ii) [pic] and [pic] are differentiable on ( a, b ).
Then there is at least one points [pic] such that
[pic].
Proof Let [pic] , [pic].
[pic] (i) [pic] is continuous on [ a, b ].
(ii) [pic] is differentiable on ( a, b ).
By Mean Value Theorem, [pic] such that [pic], hence the result is obtained. ( Why ? )
Remark: Suppose that [pic] and [pic] are differentiable on ( a, b ) and that [pic], [pic] then [pic].
This is useful to establish an inequality by using generalized mean value theorem.
Example (a) Let [pic] and [pic] be real-valued functions continuous on [pic] and differentiable in [pic].
(i) By considering the function
[pic] on [pic], or otherwise,
show that there is [pic] such that [pic]
(ii) Suppose [pic] for all [pic]. Show that [pic] for any[pic].
If, in addition, [pic] is increasing on [pic], show that [pic] is also
Increasing on [pic].
(b) Let [pic]
Show that [pic] is continuous at [pic] and increasing on [pic].
Hence or otherwise, deduce that for [pic], [pic].
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