Advanced Level Pure Mathematics

Derivatives

Advanced Level Pure Mathematics

Calculus I

Chapter 4Derivatives

4.1Introduction2

4.2Differentiability4

Continuity and Differentiability5

4.3Rules of Differentiation6

4.5Higher Derivatives9

4.6Mean Value Theorem13

4.1Introduction

Let be a fixed point and be a variable point on the curve as shown on about figure. Then the slope of the line AP is given by or . When the variable point P moves closer and closer to A along the curve , i.e. . 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 . This term is defined to be the derivative of at and is usually denoted by . The definition of derivative at any point x may be defined as follows.

DefinitionLet be a function defined on the interval and .

is said to be differentiable at ( or have a derivative at ) if the limit exists. This lime value is denoted by or and is called the derivative of at .

If has a derivative at every point x in , then is said to be differentiable on .

RemarkAs , the difference between x and is very small, i.e. tends to zero. Usually, this difference is denoted by h or . Then the derivative at may be rewritten as . ( First Principle )

ExampleLet . Find .

HF p.138 (4.1)

ExampleLet . Find .

ExampleIf , find .

HF p.139 (4.4)

ExampleLet f be a real-valued function defined on R such that for all real numbers x and y, . Suppose f is differentiable at , where .

(a)Find the value of .

(b)Show that f is differentiable on R and express in terms of .

HF p.140 (4.6)

4.2Differentiability

ExampleLet . Show that , f is continuous but not differentiable.

Solution

By definition,==

Since does not exist, is not differentiable at .

ExampleIf , show that , f is continuous but not differentiable.

HF p.142 (4.8)

ExampleA function is defined as .

Find a, b ( in term of c ) if exists.

BR p.377 EX9.1 (1)modi

Continuity and Differentiability

TheoremLet . If exist, then implies that .

ProofSince , we have

( Since both limit exist )

As exists, is bounded. Futhermore, and so .

TheoremIf is differentiable at then is continuous at .

ProofBy Theorem,, i.e..

HF p.148 (thm4.2)

Hence, is continuous at .

RemarkWe should have a clear concept about the difference between

(a) is well-defined at .

(b)the limit of at exists.

(d) is differentiable at .

DCL

ExampleProve that if satisfy and where , then exist and . Find

HF p.149 EX4A (9)

4.3Rules of Differentiation

Composite functions

Algebraic functions

where k must be independent of x(usually a constant)

Inverse functions (esp.: inverse of trigo func)

Trigonometric functions

Logarithmic functions

Parametric functions (commonly use in Rate of change)

TheoremChain Rule

If , i.e. and f, g are differentiable, then .

Example, =

ExampleFind the derivatives of the following functions:

(a)(b)

HF p.155 (4.15)

(c)(d), where

HF p.156 (4.16)

Example( Derivatives of inverse function )

Prove

SolutionLet .=

ExampleProve=

Solution

Remark=,=

=,=

Example*(a)Find(b)Find

BR p.41 (for4.1, for4.3)

ExampleFind if

(a), where a is a constant.

(b).

HF p.162 (4.24)

ExampleIf and , find

4.5Higher Derivatives

DefinitionIf is a function of , then the nth derivatives of y w.r.t. x is defined as if is differentiable.

Symbolically, the nth derivatives of y w.r.t. x is denoted by or .

Remark1. but

2.If is function of , , then (.).

ExampleLet . Find .

HF p.170(4.28)

ExampleProve that satisfies the equation .

ExampleFind a general formula for the th derivative of

(a)()

(b)

(c)

TheoremLet and be two functions which are both differentiable up to nth order. Then

(a)

(b)

TheoremLeibniz's Theorem

Let and be two functions with nth derivative. Then

where .

ExampleLet , where a is a real constant. Find .

HF p.175 (4.33)

ExampleLet . Show that for , .

HF p. 176 (4.35)

4.6Mean Value Theorem

DefinitionLet be a function defined on an intervalI. f is said to have an absolute maximum at c if I and is called the absolute maximum value.

Similarly, f is said to have an absolute minimum at d if I and is called the absolute minimum value.

TheoremFermat's Theorem

Let be defined and differentiable on an open interval (a, b). If attains its absolute maximum or absolute minimum (both are called absolute extremum) at , where , then .

ProofFor any there exists a real number h such that and . Now, suppose attains its absolute maximum at . Then we have and , and so and . Now, the left and right hand derivatives are given by

,( since )

and.( since )

Since is differentiable at , the left and right hand derivatives must be equal,

i.e. . This is possible only if .

The proof for attaining its absolute minimum at is similar and is left as an exercise.

Remark1.NOT IMPLIES absolute max. or min. at .

e.g. at , not max. and min.

figure

2.Fermat's Theorem can't apply to function in closed interval.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.defined on [ 0, 5] attains its absolute max. at but its right hand derivative does not exist.

3.Fermat' s Theorem can't apply to function which are not differentiable.

e.g. . Not differentiable at but min. at .

figure

Theorem Rolle's Theorem

If a function satisfies all the following three conditions:

(1) is continuous on the closed interval ,

(2) is differentiable in the open interval ,

(3);

then there exists at least a point such that .

ProofSince is continuous on is bounded

(i), where m (min), M (Max) are constant.

(ii), the max. and min. cannot both occur at the end points a, b.

such that

i.e. sufficiently closed to p.

By Fermat's Theorem, exist and equal to 0.

ExampleDefine on [0,4]. Note that .

We have and so . Since , 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 and is horizontal (i.e. parallel to the x-axis) then there must be at least a point (or more than one point) lying between a and b such that the tangent at this point is horizontal.

TheoremMean Value Theorem

If a function is

(1)continuous on the closed interval and

(2)differentiable in the open interval ,

then there exists at least a point such that

ProofConsider the function g defined by

is differentiable and continuous on .

Let

is also differentiable and continuous on .

We have

By Rolle's Theorem, such that

Remark:1.The Mean Value Theorem still holds for . .

2.Another form of Mean Value Theorem

3.The value of p can be expressed as , .

ExampleUse the Mean Value Theorem. show

(a)

(b),.

HF p.189 EX 4E(1a, b)

ExampleBy using Mean Value Theorem, show that

for all real values and .

SolutionLet .

Case (i)

HF p.186(4.42)

Case (ii)

Case (iii)

ExampleLet such that and be a differentiable function on such that , and is strictly decreasing. Show that .

HF p.187 (4.43)

ExampleLet be a continuous function defined on [ 3, 6 ]. If is differentiable on ( 3, 6 ) and . Show that .

HF p.189 EX4E (2)

ExampleLet be a real-valued function defined on . If is an increasing function,

show that

HF p.190 EX4E (6)

ExampleLet be a real-valued function such that

Show that is a differentiable function.

Hence deduce that for all , where is a real constant.

HF p.190 EX4E (9)

ExampleLet be a function such that is strictly increasing for .

(a)Using Mean Value Theorem, or otherwise, show that

(b)Hence, deduce that

HF p.193 RE (10)

TheoremGeneralized Mean Value Theorem

Let and such that

(i) and are continuous on [ a, b ].

(ii) and are differentiable on ( a, b ).

Then there is at least one points such that

ProofLet , .

(i) is continuous on [ a, b ].

(ii) is differentiable on ( a, b ).

By Mean Value Theorem, such that , hence the result is obtained. ( Why ? )

Remark:Suppose that and are differentiable on ( a, b ) and that , then .

This is useful to establish an inequality by using generalized mean value theorem.

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