{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# Derivatives - Derivatives Advanced Level Pure Mathematics...

This preview shows pages 1–6. Sign up to view the full content.

Derivatives Advanced Level Pure Mathematics 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 10 4.5 Higher Derivatives 17 4.6 Mean Value Theorem 22 Prepared by K. F. Ngai(2003) Page 1 4

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Derivatives Advanced Level Pure Mathematics 4.1 INTRODUCTION Let ) , ( 0 0 y x A be a fixed point and ) , ( y x P be a variable point on the curve ) ( x f y = as shown on about figure. Then the slope of the line AP is given by 0 0 x x y y - - or 0 0 ) ( ) ( x x x f x f - - . When the variable point P moves closer and closer to A along the curve ) ( x f y = , i.e. 0 x x . 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 0 0 ) ( ) ( lim 0 x x x f x f x x - - . This term is defined to be the derivative of ) ( x f at 0 x x = and is usually denoted by ) ( ' 0 x f . The definition of derivative at any point x may be defined as follows. Definition Let ) ( x f y = be a function defined on the interval [ ] b a , and ( 29 b a x , 0 . ) ( x f is said to be differentiable at 0 x ( or have a derivative at 0 x ) if the limit 0 0 ) ( ) ( lim 0 x x x f x f x x - - exists. This lime value is denoted by ) ( ' 0 x f or 0 x x dx dy = and is called the derivative of ) ( x f at 0 x . If ) ( x f has a derivative at every point x in ( 29 b a , , then ) ( x f is said to be differentiable on ( 29 b a , . Remark As 0 x x , the difference between x and 0 x is very small, i.e. 0 x x - tends to zero. Usually, this difference is denoted by h or x . Then the derivative at 0 x may be rewritten as h x f h x f h ) ( ) ( lim 0 0 0 - + . ( First Principle ) Example Let 1 ) ( 2 + = x x f . Find ) 2 ( ' f . Prepared by K. F. Ngai(2003) Page 2
Derivatives Advanced Level Pure Mathematics Example Let = = 0 , 0 0 , 1 cos ) ( 2 x x x x x f . Find ) 0 ( ' f . Example If x x f ln ) ( = , find (x) f ' . Example Let f be a real-valued function defined on R such that for all real numbers x and y, ) ( ) ( ) ( y f x f y x f + = + . Suppose f is differentiable at 0 x , where R x 0 . (a) Find the value of h h f h ) ( lim 0 . (b) Show that f is differentiable on R and express (x) f ' in terms of 0 x . Prepared by K. F. Ngai(2003) Page 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Derivatives Advanced Level Pure Mathematics 4.2 DIFFERENTIABILITY Example Let = = 0 0, 0 , 1 sin ) ( x x x x x f . Show that 0 = x , f is continuous but not differentiable. Solution By definition, (0) ' f = h f h f h ) 0 ( ) 0 ( lim 0 - + = h ) f( h f h 0 ) ( lim 0 - = h h h h 1 sin lim 0 = h h 1 sin lim 0 . Since h h 1 sin lim 0 does not exist, f is not differentiable at 0 = x . Example If 3 1 ) ( x x f = , show that 0 = x , f is continuous but not differentiable. Example Let - = 1 x if , 1 2 1 if , ) ( 2 x x x x f . Find ) ( f ' 1 + and ) ( f ' 1 - by definition. Does (1) ' f exist? Why? Prepared by K. F. Ngai(2003) Page 4
Derivatives Advanced Level Pure Mathematics Example Show that ) ( x x f = is not differentiable at 0 = x . Find also the derivative of ) ( x f when 0 x . Example Let < + = , 0 0 ) ( 2 x x x x x x f Find (0) ' f .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 28

Derivatives - Derivatives Advanced Level Pure Mathematics...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online