Derivatives Advanced Level Pure Mathematics Advanced Level Pure Mathematics 4 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 Prepared by K. F. Ngai Page 1 Derivatives 4.1 Advanced Level Pure Mathematics INTRODUCTION Let A( x0 , y0 ) be a fixed point and P ( x, y ) be a variable point on the curve y f (x) as shown on about f ( x) f ( x 0 ) y y0 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 figure. Then the slope of the line AP is given by f ( x) f ( x 0 ) . x x0 This term is defined to be the derivative of f (x) at x x0 and is usually denoted by f ' ( x 0 ) . The the curve at the point A. Hence, the slope of the tangent line at the point A is equal to lim x x0 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 x0 ( or have a derivative at x0 ) if the limit lim x x0 f ( x) f ( x 0 ) dy exists. This lime value is denoted by f ' ( x 0 ) or dx x x0 and is called the x x0 derivative of f (x) at x0 . 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 x0 is very small, i.e. x x0 tends to zero. Usually, this difference is denoted by h or x . Then the derivative at x0 may be rewritten as lim h0 Example f ( x0 h) f ( x0 ) . ( First Principle ) h Let f ( x) x 2 1 . Find f ' (2) . Prepared by K. F. Ngai Page 2 Derivatives Advanced Level Pure Mathematics Example 1 2 x cos , x 0 Let f ( x) . Find f ' (0) . x 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 x0 , where x0 R . (a) (b) f ( h) . h 0 h Show that f is differentiable on R and express f ' (x) in terms of x0 . Find the value of lim Prepared by K. F. Ngai Page 3 Derivatives 4.2 Example Advanced Level Pure Mathematics DIFFERENTIABILITY 1 x sin , x 0 Let f ( x) . Show that x 0 , f is continuous but not differentiable. x 0, x0 Solution By definition, f ' (0) = lim h 0 f (0 h) f (0) h = lim h 0 f (h) f( 0 ) h h sin Since lim sin h 0 = lim h 0 h = lim sin 1 . h h 0 1 h 1 does not exist, f is not differentiable at x 0 . h 1 Example If f ( x) x 3 , show that x 0 , f is continuous but not differentiable. Example x2 , if x 1 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 Page 4 Derivatives Example Advanced Level Pure Mathematics Show that f ( x) x is not differentiable at x 0 . Find also the derivative of f (x) when x 0. Example x 2 x x 0 Let f ( x) , Find f ' (0) . x0 x Example Let f ( x) x 3 . Find f ' (0) . Prepared by K. F. Ngai Page 5 Derivatives Example Advanced Level Pure Mathematics x2 ,x c 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 . Proof Since y ( y )x , we have lim y x 0 x ( y ) x x y lim lim x x0 x x0 ( 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 . x 0 Theorem If y f (x) is differentiable at x0 , then f (x) is continuous at x0 . Proof By Theorem, lim y 0 , i.e. lim [ f ( x0 x) f ( x0 )] 0 . x 0 x 0 x 0 Hence, f (x) is continuous at x 0 . Prepared by K. F. Ngai Page 6 Derivatives Remark Advanced Level Pure Mathematics 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 Example C L x2 a2 ,0 x a a Let f ( x) 0 ,x a . 2 2 a( x a ) ,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 Page 7 Derivatives Example Advanced Level Pure Mathematics Prove that if f satisfies f ( xy) f ( x) f ( y ) , then f '(x) f '( 1 ) , where x 0 and x find f ' ( x) . Example 1 x Find a, b in terms of c for f ' (c ) exists where f ( x) a bx 2 Example Let f be a real-valued function such that f ( x) f ( a ) ( x a ) 2 if x c, c 0 if x c (x, a R) Show that f ' ( x) 0 for all x R . Prepared by K. F. Ngai Page 8 Derivatives 4.3 Advanced Level Pure Mathematics RULES OF DIFFERENTIATION Composite functions d du ku k dx dx v u dx v d du u dx d dv du uv u v dx dx dx dy dy du dx du dx dy 1 dx dx dy dv dx v d du dv (u v) dx dx dx 2 Algebraic functions d k x kx k 1 dx where k must be independent of x (usually a constant) Inverse functions (esp.: inverse of trigo func) If y f 1 ( x ) then dy 1 df ( y) dx dy Trigonometric functions d sin x cos x dx d cos x sin x dx d tan x sec 2 x dx d sec x sec x tan x dx d csc x csc x cot x dx d cot x csc2 x dx Logarithmic functions d x e ex dx d 1 ln x dx x d x a a x ln a dx d 1 log a x dx x ln a Parametric functions (commonly use in Rate of change) dy dy dt dx dx dt Prepared by K. F. Ngai Page 9 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 ) = = = Example Find the derivatives of the following functions: (a) y x ln( 1 x) (c) y (3x 100) Example sec 2 e sin x (e sin x cos x) (b) 60 (d) y ye x 1 x2 sin 1 x , where x 0 ( Derivatives of inverse ) Let sin( xy) xy, find dy . dx Prepared by K. F. Ngai Page 10 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 . Example x = sin y 1 = cos y dy = dx 1 cos y d (sin 1 x) = dx 1 d cos 1 x dx Prove dy dx 1 x2 -1 = 1 x2 Solution Remark d dx d ( s in 1 x ) 1 ta n 1 x dx d se c 1 x dx Example* 1 = (a) Find = x 1 1 x d ln x dx x 1 cos x dx d , 1x = d , 1 cot 1 x = x = dx d , csc dx (b) Find -1 = 1 x -1 1x -1 x x 1 d x e dx Prepared by K. F. Ngai Page 11 Derivatives Example Advanced Level Pure Mathematics Find (a) (b) dy if dx y a x 1 , where a is a constant. 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 Example If x cos t and y sin t , find dy dx Exercise 4B 4(c), (d), (i), (k), (n), (q), (n), (v), (w), (x), (y), (z) Prepared by K. F. Ngai Page 12 Derivatives 4.5 Advanced Level Pure Mathematics 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 if is differentiable. dx n 1 dx n dx dx n1 Symbolically, the nth derivatives of y w.r.t. x is denoted by y ( n ) , f ( n ) ( x) or Remark 1. dy dx 1 dx dy but dny . dx n d2y 1 2 dx d 2x dy 2 d 2 y d dy dx 2 dx dx 2. If y is function of z , z f (x) , then d 2 y d dy = dx 2 dx dx dy dz dy dy dz f ' ( x) ). (. dx dz dx dx dx dy d dy f ' ( x) ( ) dz dx dz = f ' ' ( x) = f ''(x) = 2 dy 2 d y f ''(x) ( f '(x)) dz dz 2 dy dz d dy f '(x) dz dx dz dz d2y . dx 2 Example Let y sin 1 x . Find 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 Page 13 Derivatives Example Advanced Level Pure Mathematics If y xu , where u is a function of x, Prove y ( n) xu ( n) nu ( n 1) where y (n ) and u ( n ) are the nth derivative of y and u respectively. n 1 1 x Hence, if y x e , prove y Example (n) (1) x n ( n 1) 1 x e . 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 Page 14 Derivatives Theorem Theorem Advanced Level Pure Mathematics Let f (x) and g (x ) be two functions which are both differentiable up to nth order. Then (a) dn dn kf ( x ) k f ( x) dx n dx n (b) dn dn dn [ f ( x ) g ( x )] f ( x ) g ( x) dx n dx n dx n Leibniz's Theorem Let f and g be two functions with nth derivative. Then n dn [ f ( x ) g ( x )] Crn f ( r ) ( x)g ( nr ) ( x) where f ( 0) ( x) f ( x) . n dx r 0 Example Let y xeax , where a is a real constant. Find y ( 20) . Example Find Example Let y tan 1 x . Show that for n 1 , (1 x 2 ) y ( n 2) 2(n 1) xy( n1) n(n 1) y n 0 . dn 3 ( x ln x) , x 0 . dx n Prepared by K. F. Ngai Page 15 Derivatives Example Advanced Level Pure Mathematics Prove that if y x 2 cos x, then x 2 2 d y dy 4 x ( x 2 6) y 0. 2 dx dx dny d n2 y Deduce that when x 0 (n 2)( n 3) n n(n 1) n2 0 dx dx Example (a) Prove if y e x , y ( n2) 2 xy( n1) 2(n 1) y ( n ) 0 . 2 d n x2 d2 d x2 (b) Show if (e ) e f ( x) , then f ( x) 2 x f ( x) 2nf ( x) 0 . n 2 dx dx dx Prepared by K. F. Ngai Page 16 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) lim h 0 f(c h) f(c) 0, h ( since f(c h) f(c) 0 ) f(c h) f(c) 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, and f' (c) lim 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. 2. 3. figure 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. 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 Page 17 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] (i) f (x ) is bounded 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 Page 18 Derivatives Theorem Advanced Level Pure Mathematics 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 ) ( x a ) f (a ) g (x ) is differentiable and continuous on (a, b) . ba Let h( x) g ( x) f ( x) g ( x) h(x) is also differentiable and continuous on (a, b) . We have h(a ) 0, h(b) 0. (a, b) such that h ' ( ) 0 By Rolle's Theorem, g ' ( ) f ' ( ) 0 Remark: 1. 2. 3. Example f ' ( ) f (b) f (a ) ba f (b) f (a) . ba Another form of Mean Value Theorem f (b) f (a) f ' ( p)(b a) The Mean Value Theorem still holds for a b . f ' ( p ) The value of p can be expressed as p a (b a) , 0 1. f (b) f (a) f ' (a (b a))(b a) Use the Mean Value Theorem. show a b (a) sin a sin b a b (b) cos ax cos bx a b , x 0 x (c) sin px p, x p 0, x 0 . Prepared by K. F. Ngai Page 19 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) Case (ii) Case (iii) 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 Page 20 Derivatives Example Advanced Level Pure Mathematics Let P( x) a n x n a n 1 x n 1 a0 be a polynomial with real coefficients. an a 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. If Example Let f be a real-valued function defined on (0, ) . If f ' (t ) is an increasing function, f (n) f ' (n)(t n) f (t ) f (n) f ' (n 1)(t n) (t (n, n 1)) show that 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 Page 21 Derivatives Example Advanced Level Pure Mathematics 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) (b) Hence, deduce that f ' (1) f ' (2) f ' (n 1) f (n) f (1) f ' (2) f ' (3) f ' (n) 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 ). By Mean Value Theorem, p (a, b) such that h ' ( p ) h(b) h(a ) , 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 , x (a, b) then f (b) f (a) f ' ( p) . g (b) g (a) g ' ( p) This is useful to establish an inequality by using generalized mean value theorem. Prepared by K. F. Ngai Page 22 Derivatives Example Advanced Level Pure Mathematics (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) . If, in addition, f ( x) f (a ) f ' ( x) is increasing on (a, b) , show that P( x) is also g ( x) g (a ) g ' ( x) Increasing on (a, b) . e x cos x 1 if sin x cos x 1 (b) Let Q( x) 1 if x 0, 4 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 Page 23