ࡱ> prklmno[9 RTTbjbjrOVl\\\\TMTMTM8MDO,@Qp$p$p$pwww-//////$ PSwwwwwS\\$p$phw$\l$p$p-w-`,$pP N=RLKTMxT~0^E}t^\\\\ 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  EMBED Equation.3  be a fixed point and  EMBED Equation.3  be a variable point on the curve  EMBED Equation.3  as shown on about figure. Then the slope of the line AP is given by  EMBED Equation.3  or  EMBED Equation.3 . When the variable point P moves closer and closer to A along the curve  EMBED Equation.3 , i.e.  EMBED Equation.3 . 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  EMBED Equation.3 . This term is defined to be the derivative of  EMBED Equation.3  at  EMBED Equation.3  and is usually denoted by  EMBED Equation.3 . The definition of derivative at any point x may be defined as follows. Definition Let  EMBED Equation.3  be a function defined on the interval  EMBED Equation.3  and  EMBED Equation.3 .  EMBED Equation.3  is said to be differentiable at  EMBED Equation.3  ( or have a derivative at  EMBED Equation.3  ) if the limit  EMBED Equation.3  exists. This lime value is denoted by  EMBED Equation.3  or  EMBED Equation.3  and is called the derivative of  EMBED Equation.3  at  EMBED Equation.3 . If  EMBED Equation.3  has a derivative at every point x in  EMBED Equation.3 , then  EMBED Equation.3  is said to be differentiable on  EMBED Equation.3 . Remark As  EMBED Equation.3 , the difference between x and  EMBED Equation.3  is very small, i.e.  EMBED Equation.3  tends to zero. Usually, this difference is denoted by h or  EMBED Equation.3 . Then the derivative at  EMBED Equation.3  may be rewritten as  EMBED Equation.3 . ( First Principle ) Example Let  EMBED Equation.3 . Find  EMBED Equation.3 . Example Let  EMBED Equation.3 . Find  EMBED Equation.3 . Example If  EMBED Equation.3 , find  EMBED Equation.3  . Example Let f be a real-valued function defined on R such that for all real numbers x and y,  EMBED Equation.3 . Suppose f is differentiable at  EMBED Equation.3 , where  EMBED Equation.3 . (a) Find the value of  EMBED Equation.3 . (b) Show that f is differentiable on R and express  EMBED Equation.3  in terms of  EMBED Equation.3 . 4.2 Differentiability Example Let  EMBED Equation.3 . Show that  EMBED Equation.3 , f is continuous but not differentiable. Solution By definition,  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3 . Since  EMBED Equation.3  does not exist,  EMBED Equation.3  is not differentiable at  EMBED Equation.3 . Example If  EMBED Equation.3 , show that  EMBED Equation.3 , f is continuous but not differentiable. Example Let  EMBED Equation.3 . Find  EMBED Equation.3  and  EMBED Equation.3  by definition. Does  EMBED Equation.3  exist? Why? Example Show that  EMBED Equation.3  is not differentiable at  EMBED Equation.3 . Find also the derivative of  EMBED Equation.3  when  EMBED Equation.3 . Example Let  EMBED Equation.3  Find  EMBED Equation.3 . Example Let  EMBED Equation.3 . Find  EMBED Equation.3 . Example A function  EMBED Equation.3  is defined as  EMBED Equation.3 . Find a, b ( in term of c ) if  EMBED Equation.3  exists. Continuity and Differentiability  Theorem Let  EMBED Equation.3 . If  EMBED Equation.3  exist, then  EMBED Equation.3  implies that  EMBED Equation.3 . Proof Since  EMBED Equation.3 , we have EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  ( Since both limit exist )  EMBED Equation.3 . As  EMBED Equation.3  exists,  EMBED Equation.3  is bounded. Futhermore,  EMBED Equation.3  and so  EMBED Equation.3 . Theorem If  EMBED Equation.3  is differentiable at  EMBED Equation.3  then  EMBED Equation.3  is continuous at  EMBED Equation.3  . Proof By Theorem,  EMBED Equation.3 , i.e. EMBED Equation.3 . Hence,  EMBED Equation.3  is continuous at  EMBED Equation.3 . Remark We should have a clear concept about the difference between (a)  EMBED Equation.3  is well-defined at  EMBED Equation.3 . (b) the limit of  EMBED Equation.3  at  EMBED Equation.3  exists. (c)  EMBED Equation.3  is continuous at  EMBED Equation.3 . (d)  EMBED Equation.3  is differentiable at  EMBED Equation.3 . D  EMBED Equation.3  C  EMBED Equation.3  L Example Let  EMBED Equation.3 . Show  EMBED Equation.3  is continuous at  EMBED Equation.3  and discuss the continuity of  EMBED Equation.3  at  EMBED Equation.3 . Example Prove that if  EMBED Equation.3  satisfy  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3  where  EMBED Equation.3 , then  EMBED Equation.3  exist  EMBED Equation.3  and  EMBED Equation.3 . Find  EMBED Equation.3  Example Prove that if  EMBED Equation.3  satisfies  EMBED Equation.3 , then  EMBED Equation.3 , where  EMBED Equation.3  and find  EMBED Equation.3 . Example Find a, b in terms of c for  EMBED Equation.3  exists where  EMBED Equation.3  Example Let  EMBED Equation.3  be a real-valued function such that  EMBED Equation.3   EMBED Equation.3  Show that  EMBED Equation.3  for all  EMBED Equation.3 . 4.3 Rules of Differentiation Composite functions EMBED Equation EMBED Equation EMBED Equation EMBED Equation  EMBED Equation  EMBED Equation  Algebraic functions EMBED Equation  where k must be independent of x (usually a constant) Inverse functions (esp.: inverse of trigo func) EMBED Equation  Trigonometric functions EMBED Equation EMBED Equation EMBED Equation  EMBED Equation  EMBED Equation  EMBED Equation  Logarithmic functions EMBED Equation EMBED Equation  EMBED Equation  EMBED Equation  Parametric functions (commonly use in Rate of change) EMBED Equation  Theorem Chain Rule If  EMBED Equation.3  , i.e.  EMBED Equation.3  and f, g are differentiable, then  EMBED Equation.3 . Example  EMBED Equation.3 ,  EMBED Equation.3  = = =  EMBED Equation.3  Example Find the derivatives of the following functions: (a)  EMBED Equation.3  (b)  EMBED Equation.3  (c) EMBED Equation.3  (d)  EMBED Equation.3 , where  EMBED Equation.3  Example ( Derivatives of inverse ) Let  EMBED Equation.3  find  EMBED Equation.3 . Example ( Derivatives of inverse function ) Prove  EMBED Equation.3  Solution Let  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  Example Prove  EMBED Equation.3  =  EMBED Equation.3  Solution Remark  EMBED Equation.3  =  EMBED Equation.3  ,  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  ,  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  EMBED Equation.3  ,  EMBED Equation.3  =  EMBED Equation.3  Example* (a) Find  EMBED Equation.3  (b) Find  EMBED Equation.3  Example Find  EMBED Equation.3  if (a)  EMBED Equation.3 , where a is a constant. (b)  EMBED Equation.3 . Example Find the derivative of following functions (a)  EMBED Equation.3  (b)  EMBED Equation.3  Example If  EMBED Equation.3  and  EMBED Equation.3 , find  EMBED Equation.3  Exercise 4B 4(c), (d), (i), (k), (n), (q), (n), (v), (w), (x), (y), (z) 4.5 Higher Derivatives Definition If  EMBED Equation.3  is a function of  EMBED Equation.3 , then the nth derivatives of y w.r.t. x is defined as  EMBED Equation.3  if  EMBED Equation.3  is differentiable. Symbolically, the nth derivatives of y w.r.t. x is denoted by  EMBED Equation.3  or  EMBED Equation.3 . Remark 1.  EMBED Equation.3  EMBED Equation.3  but  EMBED Equation.3   EMBED Equation.3  2. If  EMBED Equation.3  is function of  EMBED Equation.3 ,  EMBED Equation.3 , then  EMBED Equation.3  (. EMBED Equation.3 ).  EMBED Equation.3 = EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Example Let  EMBED Equation.3 . Find  EMBED Equation.3 . Example Prove that  EMBED Equation.3  satisfies the equation  EMBED Equation.3 . Example If  EMBED Equation.3 , where u is a function of x, Prove  EMBED Equation.3  where  EMBED Equation.3  and  EMBED Equation.3  are the nth derivative of y and u respectively. Hence, if  EMBED Equation.3 , prove  EMBED Equation.3 . Example Find a general formula for the  EMBED Equation.3 th derivative of (a)  EMBED Equation.3  ( EMBED Equation.3 ) (b)  EMBED Equation.3  (c)  EMBED Equation.3   EMBED Equation.3  Theorem Let  EMBED Equation.3  and  EMBED Equation.3  be two functions which are both differentiable up to nth order. Then (a)  EMBED Equation.3  (b)  EMBED Equation.3  Theorem Leibniz's Theorem Let  EMBED Equation.3  and  EMBED Equation.3  be two functions with nth derivative. Then  EMBED Equation.3  where  EMBED Equation.3 . Example Let  EMBED Equation.3 , where a is a real constant. Find  EMBED Equation.3 . Example Find  EMBED Equation.3 ,  EMBED Equation.3 . Example Let  EMBED Equation.3 . Show that for  EMBED Equation.3 ,  EMBED Equation.3 . Example Prove that if  EMBED Equation.3  then  EMBED Equation.3  Deduce that when  EMBED Equation.3   EMBED Equation.3  Example (a) Prove if  EMBED Equation.3  ,  EMBED Equation.3 . (b) Show if  EMBED Equation.3 , then  EMBED Equation.3 . 4.6 Mean Value Theorem Definition Let  EMBED Equation.3  be a function defined on an interval I. f is said to have an absolute maximum at c if  EMBED Equation.3 I and  EMBED Equation.3  is called the absolute maximum value. Similarly, f is said to have an absolute minimum at d if  EMBED Equation.3 I and  EMBED Equation.3  is called the absolute minimum value. Theorem Fermat's Theorem Let  EMBED Equation.3  be defined and differentiable on an open interval (a, b). If  EMBED Equation.3  attains its absolute maximum or absolute minimum (both are called absolute extremum) at  EMBED Equation.3 , where  EMBED Equation.3 , then  EMBED Equation.3 . Proof For any  EMBED Equation.3  there exists a real number h such that  EMBED Equation.3  and  EMBED Equation.3 . Now, suppose  EMBED Equation.3  attains its absolute maximum at  EMBED Equation.3 . Then we have  EMBED Equation.3  and  EMBED Equation.3  , and so  EMBED Equation.3  and  EMBED Equation.3 . Now, the left and right hand derivatives are given by  EMBED Equation.3 , ( since  EMBED Equation.3  ) and  EMBED Equation.3 . ( since  EMBED Equation.3  ) Since  EMBED Equation.3  is differentiable at  EMBED Equation.3 , the left and right hand derivatives must be equal, i.e.  EMBED Equation.3 . This is possible only if  EMBED Equation.3 . The proof for  EMBED Equation.3  attaining its absolute minimum at  EMBED Equation.3  is similar and is left as an exercise. Remark 1.  EMBED Equation.3  NOT IMPLIES absolute max. or min. at  EMBED Equation.3 . e.g.  EMBED Equation.3  at  EMBED Equation.3 , not max. and min. figure 2. Fermat's Theorem can't apply to function in closed interval. EMBED Equation.3 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.  EMBED Equation.3 defined on [ 0, 5] attains its absolute max. at  EMBED Equation.3  but its right hand derivative does not exist. 3. Fermat' s Theorem can't apply to function which are not differentiable. e.g.  EMBED Equation.3  . Not differentiable at  EMBED Equation.3  but min. at  EMBED Equation.3 . figure Theorem Rolle's Theorem If a function  EMBED Equation.3  satisfies all the following three conditions: (1)  EMBED Equation.3  is continuous on the closed interval  EMBED Equation.3 , (2)  EMBED Equation.3  is differentiable in the open interval  EMBED Equation.3 , (3)  EMBED Equation.3 ; then there exists at least a point  EMBED Equation.3  such that  EMBED Equation.3 . Proof Since  EMBED Equation.3  is continuous on  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  is bounded (i)  EMBED Equation.3 , where m (min), M (Max) are constant.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  (ii)  EMBED Equation.3 , the max. and min. cannot both occur at the end points a, b.  EMBED Equation.3   EMBED Equation.3  such that  EMBED Equation.3  i.e.  EMBED Equation.3   EMBED Equation.3  sufficiently closed to p. By Fermat's Theorem,  EMBED Equation.3  exist and equal to 0. Example Define  EMBED Equation.3  on [0,4]. Note that  EMBED Equation.3 . We have  EMBED Equation.3  and so  EMBED Equation.3 . Since  EMBED Equation.3 , 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  EMBED Equation.3  and  EMBED Equation.3  is horizontal (i.e. parallel to the x-axis) then there must be at least a point  EMBED Equation.3  (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  EMBED Equation.3  is (1) continuous on the closed interval  EMBED Equation.3  and (2) differentiable in the open interval  EMBED Equation.3 , then there exists at least a point  EMBED Equation.3  such that  EMBED Equation.3 . Proof Consider the function g defined by  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  is differentiable and continuous on  EMBED Equation.3 . Let  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  is also differentiable and continuous on  EMBED Equation.3 . We have  EMBED Equation.3   EMBED Equation.3  By Rolle's Theorem,  EMBED Equation.3  such that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Remark: 1. The Mean Value Theorem still holds for  EMBED Equation.3 .  EMBED Equation.3 . 2. Another form of Mean Value Theorem  EMBED Equation.3  3. The value of p can be expressed as  EMBED Equation.3  ,  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3  Example Use the Mean Value Theorem. show  EMBED Equation.3  (a)  EMBED Equation.3  (b)  EMBED Equation.3 ,  EMBED Equation.3  (c)  EMBED Equation.3 ,  EMBED Equation.3 . Example By using Mean Value Theorem, show that  EMBED Equation.3  for all real values  EMBED Equation.3  and  EMBED Equation.3 . Solution Let  EMBED Equation.3 . Case (i) Case (ii) Case (iii) Example Let  EMBED Equation.3  such that  EMBED Equation.3  and  EMBED Equation.3  be a differentiable function on  EMBED Equation.3  such that  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3  is strictly decreasing. Show that  EMBED Equation.3 . Example Let  EMBED Equation.3  be a continuous function defined on [ 3, 6 ]. If  EMBED Equation.3  is differentiable on ( 3, 6 ) and  EMBED Equation.3 . Show that  EMBED Equation.3 . Example Let  EMBED Equation.3  be a polynomial with real coefficients. If  EMBED Equation.3  by using Mean Value Theorem, show that the equation  EMBED Equation.3  has at least one real root between 0 and 1. Example Let  EMBED Equation.3  be a real-valued function defined on  EMBED Equation.3 . If  EMBED Equation.3  is an increasing function, show that  EMBED Equation.3   EMBED Equation.3  Example Let  EMBED Equation.3  be a real-valued function such that  EMBED Equation.3 ,  EMBED Equation.3  Show that  EMBED Equation.3  is a differentiable function. Hence deduce that  EMBED Equation.3  for all  EMBED Equation.3 , where  EMBED Equation.3  is a real constant. Example Let  EMBED Equation.3  be a function such that  EMBED Equation.3  is strictly increasing for  EMBED Equation.3 . (a) Using Mean Value Theorem, or otherwise, show that  EMBED Equation.3  (b) Hence, deduce that  EMBED Equation.3  Theorem Generalized Mean Value Theorem Let  EMBED Equation.3  and  EMBED Equation.3  such that (i)  EMBED Equation.3  and  EMBED Equation.3  are continuous on [ a, b ]. (ii)  EMBED Equation.3  and  EMBED Equation.3  are differentiable on ( a, b ). Then there is at least one points  EMBED Equation.3  such that  EMBED Equation.3 . Proof Let  EMBED Equation.3  ,  EMBED Equation.3 .  EMBED Equation.3  (i)  EMBED Equation.3  is continuous on [ a, b ]. (ii)  EMBED Equation.3  is differentiable on ( a, b ). By Mean Value Theorem,  EMBED Equation.3  such that  EMBED Equation.3 , hence the result is obtained. ( Why ? ) Remark: Suppose that  EMBED Equation.3  and  EMBED Equation.3  are differentiable on ( a, b ) and that  EMBED Equation.3 ,  EMBED Equation.3  then  EMBED Equation.3 . This is useful to establish an inequality by using generalized mean value theorem. Example (a) Let  EMBED Equation.3  and  EMBED Equation.3  be real-valued functions continuous on  EMBED Equation.3  and differentiable in  EMBED Equation.3 . (i) By considering the function  EMBED Equation.3  on  EMBED Equation.3 , or otherwise, show that there is  EMBED Equation.3  such that  EMBED Equation.3  (ii) Suppose  EMBED Equation.3  for all  EMBED Equation.3 . Show that  EMBED Equation.3  for any EMBED Equation.3 . If, in addition,  EMBED Equation.3  is increasing on  EMBED Equation.3 , show that  EMBED Equation.3  is also Increasing on  EMBED Equation.3 . (b) Let  EMBED Equation.3  Show that  EMBED Equation.3  is continuous at  EMBED Equation.3  and increasing on  EMBED Equation.3 . Hence or otherwise, deduce that for  EMBED Equation.3 ,  EMBED Equation.3 . Derivatives Advanced Level Pure Mathematics Prepared by K. F. Ngai Page  PAGE 3 4 !"$-/1HIK@BN`anoŻ}vha jEHUjLm? CJKHUVo( jEHUjEm? CJKHUVo( jUjCJUmHnHtH u 5;CJ\ 5CJ\CJ5CJ(OJQJ\^J5;CJ(OJQJ5CJPOJQJ^J5CJPOJQJj5CJPOJQJU5;CJ4OJQJ5CJ4OJQJ5CJ4OJQJo( !#$/01IJnABCDEWd1$7$8$9D]W^`Wd1$7$8$9D]W V1$7$8$9D]VVd1$7$8$9D]V 1$7$8$9D`1$7$8$9DSLTSTEFGHIJKLMN`bcdefghijWX1 $^a$ $^`a$ 9r !#01DEFGKL_`ab|}!"VWdexyz{y jgEHUjvm? CJKHUVo( jn EHUjsm? CJKHUVo( jl EHUjmm? CJKHUVo( j EHUjdm? CJKHUVo( jEHUj\m? CJKHUVo(6] jU jEHUjPm? CJKHUVo(/   Xcgh{|}~üykjm? CJKHUVo( jEHUjm? CJKHUVo( jEHUjm? CJKHUVo( jEHUjm? CJKHUVo(5\ jEHUjm? CJKHUVo( jEHUjm? CJKHUVo( jU jEHUjm? 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