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Derivatives Advanced Level Pure Mathematics Advanced Level Pure Mathematics Calculus I 4 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 Derivatives Advanced Level Pure Mathematics 4.1 INTRODUCTION Let A( x0 , y 0 ) be a fixed point and P( x, y ) be a variable point on the curve y f (x) as shown on about y y0 f ( x) f ( x 0 ) figure. Then the slope of the line AP is given by or . When the variable point P x x0 x x0 moves closer and closer to A along the curve y f (x) , i.e. x x0 . the line AP becomes the tangent line of f ( x) f ( x 0 ) the curve at the point A. Hence, the slope of the tangent line at the point A is equal to lim . x x0 x x0 This term is defined to be the derivative of f (x) at x x0 and is usually denoted by f ' ( x0 ) . The definition of derivative at any point x may be defined as follows. Definition Let y f (x) be a function defined on the interval a, b and x0 a, b . f (x) is said to be differentiable at x 0 ( or have a derivative at x 0 ) if the limit f ( x) f ( x 0 ) dy lim exists. This lime value is denoted by f ' ( x 0 ) or and is called the x x0 x x0 dx x x0 derivative of f (x) at x 0 . If f (x) has a derivative at every point x in a, b , then f (x) is said to be differentiable on a, b . Remark As x x0 , the difference between x and x 0 is very small, i.e. x x0 tends to zero. Usually, this difference is denoted by h or x . Then the derivative at x 0 may be rewritten as f ( x 0 h) f ( x 0 ) lim . ( First Principle ) h 0 h Example Let f ( x) x 2 1 . Find f ' (2) . Prepared by K. F. Ngai(2003) Page 2 Derivatives Advanced Level Pure Mathematics 2 1 x cos , x 0 Example Let f ( x) x . Find f ' (0) . 0, x0 Example If f ( x) ln x , find f ' (x) . Example Let f be a real-valued function defined on R such that for all real numbers x and y, f ( x y) f ( x) f ( y) . Suppose f is differentiable at x 0 , where x0 R . f ( h) (a) Find the value of lim . h 0 h (b) Show that f is differentiable on R and express f ' (x) in terms of x 0 . Prepared by K. F. Ngai(2003) Page 3 Derivatives Advanced Level Pure Mathematics 4.2 DIFFERENTIABILITY 1 x sin , x 0 Example Let f ( x) x . Show that x 0 , f is continuous but not differentiable. 0, x0 Solution f (0 h) f (0) f (h) f( 0 ) By definition, f ' (0) = lim = lim h 0 h h 0 h 1 h sin = lim h h 0 h 1 = lim sin . h 0 h 1 Since lim sin does not exist, f is not differentiable at x 0 . h 0 h 1 Example If f ( x ) x , show that x 0 , f is continuous but not differentiable. 3 x2 , if x 1 Example Let f ( x) . 2 x 1 , if x 1 Find f ' ( 1 ) and f ' ( 1 ) by definition. Does f ' (1) exist? Why? Prepared by K. F. Ngai(2003) Page 4 Derivatives Advanced Level Pure Mathematics Example Show that f ( x) x is not differentiable at x 0 . Find also the derivative of f (x) when x 0. x 2 x x0 Example Let f ( x) , Find f ' (0) . x x0 Example Let f ( x) x 3 . Find f ' (0) . Prepared by K. F. Ngai(2003) Page 5 Derivatives Advanced Level Pure Mathematics x2 ,x c Example A function f is defined as f ( x) . ax b , x c Find a, b ( in term of c ) if f ' (c) exists. CONTINUITY AND DIFFERENTIABILITY Theorem Let y f (x) . If f ' ( x) exist, then x 0 implies that y 0 . y y Proof Since y ( )x , we have lim y ( )x x x 0 x lim y lim x x 0 x x 0 ( Since both limit exist ) f ' ( x) lim x . x 0 As f ' ( x) exists, f ' ( x) is bounded. Futhermore, lim x 0 and so lim y 0 . x0 x 0 Theorem If y f (x) is differentiable at x0 , then f (x) is continuous at x 0 . Proof By Theorem, lim y 0 , i.e. lim [ f ( x0 x) f ( x0 )] 0 . x 0 x0 Hence, f (x) is continuous at x 0 . Prepared by K. F. Ngai(2003) Page 6 Derivatives Advanced Level Pure Mathematics Remark We should have a clear concept about the difference between (a) f (x) is well-defined at x 0 . (b) the limit of f (x) at x 0 exists. (c) f (x) is continuous at x 0 . (d) f (x) is differentiable at x 0 . D C L x2 a2 ,0 x a a Example Let f ( x) 0 ,x a . a( x a ) 2 2 ,x a x2 Show f (x) is continuous at x a and discuss the continuity of f ' ( x) at x a . Example Prove that if f satisfy f ( x y) f ( x) f ( y) x, y R and f ( x) 1 xg ( x) where lim g ( x) 1 , then f ' ( x) exist x R and f ' ( x) f ( x) . Find f (x). x 0 Prepared by K. F. Ngai(2003) Page 7 Derivatives Advanced Level Pure Mathematics f '(1 ) Example Prove that if f satisfies f ( xy) f ( x) f ( y) , then f '(x) , where x 0 and x find f (x) . 1 x if x c, c 0 Example Find a, b in terms of c for f ' (c) exists where f ( x) a bx 2 if x c Example Let f be a real-valued function such that f ( x) f (a) ( x a) 2 (x, a R) Show that f ' ( x) 0 for all x R . Prepared by K. F. Ngai(2003) Page 8 Derivatives Advanced Level Pure Mathematics Exercise 4A Name: ________________________ 1. Determine which of following functions are differentiable at x 0 . 1 x sin 2 ( x 0) (a) f ( x) x 3 (b) f ( x) x 0 ( x 0) 2. Find f ' ( x) of the following functions. Also, locate the points at which the function is not differentiable. 4x ( x 3) (a) f ( x) 2 (b) f ( x) x 3 8 2 x 6 ( x 3) x3 ,x 1 3. Let f ( x) . ax b , x 1 Find the values of a and b so that f is differentiable at x 1 . Prepared by K. F. Ngai(2003) Page 9 Derivatives Advanced Level Pure Mathematics 4.3 RULES OF DIFFERENTIATION Composite functions d du d du dv d dv du ku k (u v) uv u v dx dx dx dx dx dx dx dx du dv uv d u dy dy du dy 1 dx 2 dx dx v v dx du dx dx dx dy Algebraic functions d k x kxk 1 where k must be independent of x (usually a constant) dx Inverse functions (esp.: inverse of trigo func) dy 1 If y f 1 ( x) the n dx df ( y ) dy Trigonometric functions d d d sin x cos x cos x sin x tan x sec2 x dx dx dx d d d sec x sec x tan x csc x csc x cot x cot x csc2 x dx dx dx Logarithmic functions d x d 1 e ex ln x dx dx x d x d 1 a a x ln a log a x dx dx x ln a Parametric functions (commonly use in Rate of change) dy dy dt dx dx dt Prepared by K. F. Ngai(2003) Page 10 Derivatives Advanced Level Pure Mathematics Theorem Chain Rule If h g f , i.e. h( x) g ( f ( x)) and f, g are differentiable, then h' ( x) g ' ( f ( x)) f ' ( x) . Example h( x) tan e sin x , h' ( x ) = = = sec2 e sin x (e sin x cos x) Example Find the derivatives of the following functions: x (a) y x ln(1 x) (b) y 1 x2 1 sin (c) y (3x 100 ) 60 (d) ye x , where x 0 dy Example Let sin( xy) xy, find . dx Prepared by K. F. Ngai(2003) Page 11 Derivatives Advanced Level Pure Mathematics Example ( Derivatives of inverse function ) d 1 Prove (sin 1 x) dx 1 x2 Solution Let y sin 1 x . x = sin y dy 1 = cos y dx dy 1 = dx cos y d 1 (sin 1 x) = dx 1 x2 d -1 Example Prove cos1 x = dx 1 x2 Solution d 1 d -1 Remark (sin 1 x) = , cos1 x = dx 1 x2 dx 1 x2 d 1 d -1 tan 1 x = , cot1 x = dx 1 x2 dx 1 x2 d 1 d -1 sec1 x = , csc1 x = dx x x2 1 dx x x2 1 d d x Example* (a) Find ln x (b) Find e dx dx Prepared by K. F. Ngai(2003) Page 12 Derivatives Advanced Level Pure Mathematics dy Example Find if dx (a) y a x 1 , where a is a constant. (b) y x x sin x . Example Find the derivative of following functions (a) x 2 sin 3 (1 3x) (b) (ln (cos 2 x)) 3 dy Example If x cost and y sin t , find dx Prepared by K. F. Ngai(2003) Page 13 Derivatives Advanced Level Pure Mathematics Exercise 4B Name ________________________ 1. Find the derivatives of the following functions. (a) y e sin x (b) y tan e x (c) y ln( x 2 1) 3 sin x (d) y ln(tan e x ) (e) y sin 1 x 2x (f) y tan 1 2 (g) y (cos x) x 1 x Prepared by K. F. Ngai(2003) Page 14 Derivatives Advanced Level Pure Mathematics 1 tan (h) y 3x e x (i) y3 x 1 x (j) yx x (k) y4 ln x d 1 2. By considering the inverse of cos x , show that (cos 1 x) dx 1 x2 Prepared by K. F. Ngai(2003) Page 15 Derivatives Advanced Level Pure Mathematics dy 3. Find if dx 1 (a) tan 1 y xy x 2 (b) x 2 y 2 x cos y 1 y dy 4. Find if dx x t cost x cos3 t (a) (b) y 1 sin t y sin t 3 Prepared by K. F. Ngai(2003) Page 16 Derivatives Advanced Level Pure Mathematics 4.5 HIGHER DERIVATIVES Definition If y f (x) is a function of x , then the nth derivatives of y w.r.t. x is defined as d n y d d n1 y d n 1 y n 1 if is differentiable. dx n dx dx dx n 1 dny Symbolically, the nth derivatives of y w.r.t. x is denoted by y ( n ) , f (n) ( x) or . dx n dy 1 d2y 1 Remark 1. but 2 dx dx dx d 2x dy dy 2 d 2 y d dy dx 2 dx dx dy dy dz dy dz 2. If y is function of z , z f (x) , then (. f ' ( x) ). dx dz dx dx dx d 2 y d dy dy d dy = = f ' ' ( x) f ' ( x) ( ) dx 2 dx dx dz dx dz dy dz d dy = f ''(x) f '(x) dz dx dz dz 2 dy 2 d y = f ''(x) ( f '(x)) dz dz 2 d2y Example Let y sin 1 x . Find . dx 2 Example Prove that y e kx (a cos x b sin x) satisfies the equation y ' '2ky'(k 2 1) y 0 . Prepared by K. F. Ngai(2003) Page 17 Derivatives Advanced Level Pure Mathematics ( n 1) Example If y xu , where u is a function of x, Prove y (n) xu (n) nu where y (n ) and u (n ) are the nth derivative of y and u respectively. 1 1 Hence, if y x n 1e x , prove y ( n ) (1) n x ( n 1) e x . Example Find a general formula for the n th derivative of (a) y e ax ( a R ) (b) y sin x (c) y xm ( m R) Prepared by K. F. Ngai(2003) Page 18 Derivatives Advanced Level Pure Mathematics Theorem Let f (x) and g (x) be two functions which are both differentiable up to nth order. Then dn dn (a) kf ( x) k n f ( x) dx n dx dn dn dn (b) [ f ( x) g ( x)] n f ( x) n g ( x) dx n dx dx Theorem Leibniz's Theorem Let f and g be two functions with nth derivative. Then dn n [ f ( x) g ( x)] C rn f (r ) ( x)g ( n r ) ( x) where f (0) ( x) f ( x) . dx n r 0 Example Let y xe ax , where a is a real constant. Find y ( 20) . dn 3 Example Find ( x ln x) , x 0 . dx n Example Let y tan 1 x . Show that for n 1 , (1 x 2 ) y ( n 2) 2(n 1) xy ( n 1) n(n 1) y ( n ) 0 . Prepared by K. F. Ngai(2003) Page 19 Derivatives Advanced Level Pure Mathematics 2 d y dy Example Prove that if y x 2 cos x, then x 2 2 4 x ( x 2 6) y 0. dx dx dny d n2 y Deduce that when x 0 (n 2)( n 3) n n(n 1) n 2 0 dx dx (a) Prove if y e x , y ( n 2) 2 xy ( n 1) 2(n 1) y ( n ) 0 . 2 Example d n x2 x2 d2 d (b) Show if n (e ) e f ( x) , then 2 f ( x) 2 x f ( x) 2nf ( x) 0 . dx dx dx Prepared by K. F. Ngai(2003) Page 20 Derivatives Advanced Level Pure Mathematics Exercise 4D Name ____________________ x 1 r 1. Let r be a real number. Define y for x 1 . x 1 dy 2ry (a) Show that . dx x 2 1 (b) For n 1, 2, 3, , show that ( x 2 1) y ( n 1) 2(nx r ) y ( n ) (n 2 n) y ( n 1) 0 dky where y ( 0) y and y ( k ) for k 1 . dx k 2. Let f ( x) x n e ax , where a is real and n is a positive integer. Evaluate f ( 2 n ) (0) . Prepared by K. F. Ngai(2003) Page 21 Derivatives Advanced Level Pure Mathematics 4.6 MEAN VALUE THEOREM Definition Let y f (x) be a function defined on an interval I. f is said to have an absolute maximum at c if f (c) f ( x), x I and f (c) is called the absolute maximum value. Similarly, f is said to have an absolute minimum at d if f (d ) f ( x), x I and f(d) is called the absolute minimum value. Theorem Fermat's Theorem Let y f (x) be defined and differentiable on an open interval (a, b). If f (x) attains its absolute maximum or absolute minimum (both are called absolute extremum) at x c , where c (a, b) , then f ' (c) 0 . Proof For any x (a, b), there exists a real number h such that ( x h) (a, b) and ( x h) (a, b) . Now, suppose f (x) attains its absolute maximum at x c . Then we have (c h) (a,b) and (c h) (a,b) , and so f (c h) f (c) and f (c h) f (c) . Now, the left and right hand derivatives are given by f(c h) f(c) f'(c) lim 0, ( since f(c h) f(c) 0 ) h 0 h f(c h) f(c) and f'(c) lim 0. ( since f(c h) f(c) 0 ) h 0 h Since f (x) is differentiable at x c , the left and right hand derivatives must be equal, i.e. f ' (c) f ' (c) . This is possible only if f ' (c) 0 . The proof for f (x) attaining its absolute minimum at x c is similar and is left as an exercise. Remark 1. f ' ( p) 0 NOT IMPLIES absolute max. or min. at x p . e.g. f ( x) x 3 at x 0 , 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. f ( x) ( x 2) 2 1defined on [ 0, 5] attains its absolute max. at x 5 but its right hand derivative does not exist. 3. Fermat' s Theorem can't apply to function which are not differentiable. e.g. f ( x) x . Not differentiable at x 0 but min. at x 0 . figure Prepared by K. F. Ngai(2003) Page 22 Derivatives Advanced Level Pure Mathematics Theorem Rolle's Theorem If a function f (x) satisfies all the following three conditions: (1) f (x) is continuous on the closed interval [a, b] , (2) f (x) is differentiable in the open interval (a, b) , (3) f (a) f (b) ; then there exists at least a point (a, b) such that f ' ( ) 0 . Proof Since f (x) is continuous on [a, b] f (x) is bounded (i) m M , where m (min), M (Max) are constant. m f ( x) M , x [a, b] f ( x) M , x [a, b] f ' ( x) 0, x (a, b) (ii) m M , the max. and min. cannot both occur at the end points a, b. p (a, b) such that f ( p) M i.e. f ( p) f ( x) x sufficiently closed to p. By Fermat's Theorem, f ' ( p) exist and equal to 0. Example Define f ( x) ( x 2) 2 1 on [0,4]. Note that f (0) f (4) 5 . We have f ' ( x) 2( x 2) and so f ' (2) 0 . Since 2 (0,4) , 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 (a, f (a)) and (b, f (b)) 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. Prepared by K. F. Ngai(2003) Page 23 Derivatives Advanced Level Pure Mathematics Theorem Mean Value Theorem If a function f (x) is (1) continuous on the closed interval [a, b] and (2) differentiable in the open interval (a, b) , then there exists at least a point (a, b) such that f (b) f (a) f ' ( ) . ba Proof Consider the function g defined by f (b) f (a) g ( x) ( x a) f (a) g (x) is differentiable and continuous on (a, b) . ba Let h( x) g ( x) f ( x) h(x) is also differentiable and continuous on (a, b) . We have h(a) 0, h(b) 0. By Rolle's Theorem, (a, b) such that h ' ( ) 0 g ' ( ) f ' ( ) 0 f (b) f (a) f ' ( ) ba f (b) f (a) Remark: 1. The Mean Value Theorem still holds for a b . f ' ( p) . ba 2. Another form of Mean Value Theorem f (b) f (a) f ' ( p)(b a) 3. The value of p can be expressed as p a (b a) , 0 1. f (b) f (a) f ' (a (b a))(b a) Example Use the Mean Value Theorem. show a b (a) sin a sin b a b cos ax cos bx (b) a b , x 0 x sin px (c) p, p 0, x 0 . x Prepared by K. F. Ngai(2003) Page 24 Derivatives Advanced Level Pure Mathematics Example By using Mean Value Theorem, show that e y e a e a ( y a) for all real values y and a . Solution Let f ( x) e x . Case (i) y a Case (ii) y a Case (iii) y a Example Let a, b R such that a b and f (x) be a differentiable function on R such that f (a) 0 , f (b) 0 and f ' ( x) is strictly decreasing. Show that f ' (b) 0 . Example Let f (x) be a continuous function defined on [ 3, 6 ]. If f (x) is differentiable on ( 3, 6 ) and f ' ( x) 9 3 . Show that 18 f (6) f (3) 36 . Prepared by K. F. Ngai(2003) Page 25 Derivatives Advanced Level Pure Mathematics Example Let P ( x) a n x n a n 1 x n 1 a0 be a polynomial with real coefficients. an a If n1 a0 0, by using Mean Value Theorem, show that the equation P( x) 0 has n 1 n at least one real root between 0 and 1. Example Let f be a real-valued function defined on (0, ) . If f ' (t ) is an increasing function, show that f (n) f ' (n)(t n) f (t ) f (n) f ' (n 1)(t n) (t (n, n 1)) Example Let f be a real-valued function such that f ( x) f ( y) ( x y) 2 , (x, y R) Show that f is a differentiable function. Hence deduce that f ( x) k for all x R , where k is a real constant. Prepared by K. F. Ngai(2003) Page 26 Derivatives Advanced Level Pure Mathematics Example Let f (x) be a function such that f ' ( x) is strictly increasing for x 0 . (a) Using Mean Value Theorem, or otherwise, show that f ' (k ) f (k 1) f (k ) f ' (k 1) , k 1 (b) Hence, deduce that f ' (1) f ' (2) f ' (n 1) f (n) f (1) f ' (2) f ' (3) f ' (n) , n 2 Theorem Generalized Mean Value Theorem Let f (x) and g (x) such that (i) f (x) and g (x) are continuous on [ a, b ]. (ii) f (x) and g (x) are differentiable on ( a, b ). Then there is at least one points p (a, b) such that [ f (b) f (a)] g ' ( p) [ g (b) g (a)] f ' ( p) . Proof Let h( x) [ f (b) f (a)] g ( x) [ g (b) g (a)] f ( x) , a x b . (i) h(x) is continuous on [ a, b ]. (ii) h(x) is differentiable on ( a, b ). h(b) h(a) By Mean Value Theorem, p (a, b) such that h ' ( p) , hence the result is obtained. ba ( Why ? ) Remark: Suppose that f (x) and g (x) are differentiable on ( a, b ) and that f ' ( x) g ' ( x) 0 , f (b) f (a) f ' ( p) x (a, b) then . g (b) g (a) g ' ( p) This is useful to establish an inequality by using generalized mean value theorem. Prepared by K. F. Ngai(2003) Page 27 Derivatives Advanced Level Pure Mathematics Example (a) Let f and g be real-valued functions continuous on [a, b] and differentiable in (a, b) . (i) By considering the function h( x) f ( x)[g (b) g (a)] g ( x)[ f (b) f (a)] on [a, b] , or otherwise, show that there is c (a, b) such that f ' (c)[g (b) g (a)] g ' (c)[ f (b) f (a)] (ii) Suppose g ' ( x) 0 for all x (a, b) . Show that g ( x) g (a) 0 for any x (a, b) . f ' ( x) f ( x) f (a) If, in addition, is increasing on (a, b) , show that P ( x) is also g ' ( x) g ( x) g (a ) Increasing on (a, b) . e x cos x 1 if x 0, 4 sin x cos x 1 (b) Let Q( x) 1 if x0 Show that Q is continuous at x 0 and increasing on 0, . 4 x Hence or otherwise, deduce that for x 0, , Q (t )dt x . 4 0 Prepared by K. F. Ngai(2003) Page 28