RULES OF DIFFERENTIATION LESSON PLAN. C2 Topic Overview CALCULUS

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1 CALCULUS C Topic Overview C RULES OF DIFFERENTIATION In pracice we o no carry ou iffereniaion from fir principle (a ecribe in Topic C Inroucion o Differeniaion). Inea we ue a e of rule ha allow u o obain he erivaive of a funcion wihou he irec ue of limi. Topic C preen he baic rule of iffereniaion for polynomial funcion. LESSON PLAN Leon No. LESSON TITLE Page Eample SAQ C. Power. -. C. C. C. Conan C. Conan Muliple of Funcion C. C. A Sum (or Difference) of Funcion 9.6 C. C. Furher Eample of Differeniaion C. C.6 Furher Eample of Differeniaion.0 C.6 6 C. Higher Orer Derivaive. C. Appeni C.8 Soluion o he Self-Aemen Queion Page Topic C

2 CALCULUS C Topic Overview SUMMARY TABLE OF DIFFERENTIATION RULES RULE FUNCTION f() DERIVATIVE f'() Power n n n Conan (C) C 0 Conan muliple (a) of a funcion af() a f '() Sum of funcion f () + f () f '() + f '() Noe: The above able ummarie he rule of iffereniaion covere in Topic C only. Page Topic C

3 C. The Derivaive of a Power C. THE DERIVATIVE OF A POWER The echnique of iffereniaion from fir principle can be ue o how ha f() f '() f() f '() f() f '() f() f '() There i an obviou paern here an he general rule for iffereniaing a power i f() n f'() n n We will aume hi general rule for iffereniaing a power wihou giving a formal proof. I may be helpful o remember hi imporan rule in wor a: Muliply by he power an reuce he power by The above rule applie when n i a poiive, negaive, ecimal or fracional power. EXAMPLE. Derivaive of Poiive Power Differeniae: (i) 6 (iii) When iffereniaing a poiive power he ep o follow are imply: (a) Muliply by he power (b) Reuce he power by (i) f() 6 f() (iii) f() f '() 6 6 f '() 6 f '() 0 Noe: Reul (iii) i worh memoriing, ince occur frequenly in algebraic epreion. f () f'() Page Topic C

4 C. The Derivaive of a Power EXAMPLE. Derivaive of Negaive Power Differeniae: (i) 6 (iii) We may fir nee o re-wrie he funcion a a ingle negaive power (a hown in eample an (iii) below), hen iffereniae following he ep in he rule: (a) Muliply by he power (b) Reuce he power by (i) Simply apply he rule: f() f'() ( ) Fir, re-wrie a a ingle power of f() 6 6 Now, iffereniae uing he rule f '() ( 6) 6 Re-wrie wih a poiive power 6 6 (iii) Fir, re-wrie a a ingle power of f() Now, iffereniae uing he rule f'() ( ) Re-wrie wih a poiive power SAQ C. Differeniae he following funcion: (a) (b) (c) 0 () (e) (f) (g) (h) 9 Page Topic C

5 C. The Derivaive of a Power EXAMPLE. Derivaive of Decimal an Fracional Power Differeniae: (i). (iii) (iv) (i) Apply he iffereniaion rule for power: f(). f '().... Similarly, f() f '() (iii) Fir, change he quare-roo ign f () o a fracional power of Now, iffereniae uing he rule f '() Noe: Since original funcion i wrien uing a quare roo ign, we epre our final anwer in he ame form, ie. change he fracional ine back o a roo ign in la ep. Page Topic C

6 C. The Derivaive of a Power (iv) Fir, change he quare-roo ign o a fracional power of f () Re-wrie a a ingle power Now, iffereniae uing he rule Re-wrie uing a quare roo ign f '() SAQ C. Differeniae he following funcion: (a). (b) 0.9 (c). () (e) (f) (g) (h) Page 6 Topic C

7 C. The Derivaive of a Conan C. THE DERIVATIVE OF A CONSTANT If he funcion f() i conan C, we can fin he erivaive f '() from fir principle. [Ref: C. The Derivaive of a Funcion] give f() C f '() f( + h) f() lim h 0 h C C lim h 0 h 0 lim h 0 h lim ( 0) h 0 0 Thi how ha he erivaive of a conan i alway 0. Hence he rule for iffereniaing a conan i: f() C f '() 0 Thi rule houl be memorie. EXAMPLE. Differeniae: (i) f() 6 f(). (iii) (i) f() 6 f '() 0 f(). f '() 0 f() (iii) f() f '() 0 Page Topic C

8 C. The Derivaive of a Conan Muliple of a Funcion C. THE DERIVATIVE OF A CONSTANT MULTIPLE OF A FUNCTION If a funcion f() i of he form f() a n can ue he following rule: where a i he conan muliplier, hen we f() a n f'() an n Thi rule mean ha he conan muliplier a remain unchange uring he iffereniaion proce. Noe: Be careful no o confue hi rule wih he rule for iffereniaing a conan. A conan appearing on i own iffereniae o zero, wherea he conan muliple rule ay ha a conan which muliplie ome oher funcion remain unchange uring iffereniaion. EXAMPLE. Differeniae: (i) 6 (i) f() f() (iii) 6 f '() ( ) 6 f '() 6( ) (iii) f() f'() 8 SAQ C. Differeniae he following wih repec o : (a) (b) (c) 6 () (e) (f) Page 8 Topic C

9 C. The Derivaive of a Sum of Funcion C. THE DERIVATIVE OF A SUM (OR DIFFERENCE) OF FUNCTIONS A funcion ha i he um of wo funcion of can be iffereniae by aing ogeher he erivaive of he wo eparae funcion. The rule i: f() f () + f () f'() f '() + f '() Similarly, he rule for iffereniaing he ifference of wo funcion i: f() f () f () f'() f '() f '() The um rule can be eene o apply o any number of eparae funcion ae or ubrace ogeher a hown in he following eample. EXAMPLE.6 Differeniae: (i) (iii) (iv) + (i) f() + 6 f() + 9 f '() + 6 f'() () + () (iii) f() f' ( ) 0 + () ( 0 + ) (iv) f() f'() SAQ C. Differeniae he following wih repec o : (a) + (b) 6 8 (c) () + (e) + (f) Page 9 Topic C

10 C. Furher Eample C. FURTHER EXAMPLES OF DIFFERENTIATION The four main iffereniaion rule we have een o far are: RULE FUNCTION DERIVATIVE Power n n n Conan (C) C 0 Conan muliple (a) of a funcion af() a f'() Sum of funcion f () + f () f '() + f '() Alernaive noaion can alo be ue o repreen he erivaive [Ref: C. The Derivaive of a Funcion]. The eample below how how he noaion repreen he erivaive of y wih repec o. y i ue o EXAMPLE. y Fin in he following cae: (i) y + y (i) y + y y + y (0 9 ) + ( ) Alernaively, he following eample ue he more compac form of noaion for he erivaive of f() wih repec o. f() Page 0 Topic C

11 C. Furher Eample EXAMPLE.8 Fin he following erivaive: (i) ( ) (iii) (00) (i) ( ) ( ) () (iii) (00) 0 I i alo imporan o be able o ue he noaion an rule of iffereniaion in iuaion where he variable name may no be an y. In applicaion of calculu we are frequenly ealing wih funcion of ime, when i frequenly ue o repreen ime an he funcion i wrien a f(). EXAMPLE.9 (i) Fin f '(), if SAQ C. f'() f() k If V kq where k an k are conan, fin q (i) f() + + V kq y (a) Fin in each of he following: V q k q V q k q k k q + k k k q + q q k (q) k ( q q ) (i) y y (b) Fin he following erivaive: (iii) y (i) (u u u ) ( ) u Page Topic C

12 C.6 Furher Eample C.6 FURTHER EXAMPLES OF DIFFERENTIATION Finally, i i imporan o be able o recognie iuaion in which he given funcion canno be iffereniae irecly a i an, uing one or more of he four main rule of iffereniaion. In uch cae he funcion mu be implifie or rearrange ino a uiable form for iffereniaion. EXAMPLE.0 Differeniae he following wih repec o : (i) ( )( + ) ( + ) (iii) + (i) Fir muliply ou he bracke f() ( )( + ) hen iffereniae wih repec o f'() 0 8 Fir change he quare roo ign an muliply ou he bracke f() ( + ( + + ) hen iffereniae wih repec o f'() + ) + Page Topic C

13 C.6 Furher Eample (iii) Fir ivie all he erm on he op line by an wrie each erm of he funcion a a muliple of a ingle power of f() hen iffereniae wih repec o f'() () + ( ) + ( ) Noe: The funcion f() ( )( + ) i an eample of a prouc, ie. wo funcion + muliplie ogeher, an he funcion f() i an eample of a quoien, ie. one funcion ivie by anoher. There are wo pecial rule, calle he Prouc Rule an he Quoien Rule, which can be ue o iffereniae more complicae prouc an quoien. Thee rule are no covere in hee noe, bu can be foun in he iffereniaion echnique ecion of mo e book on calculu. To iffereniae a prouc, i i no correc ju o iffereniae he facor eparaely an hen o muliply he reul ogeher. Similarly, o iffereniae a quoien, i i no correc ju o iffereniae he op an boom line eparaely an hen o ivie one reul by he oher. SAQ C.6 y (a) Fin in each of he following: (i) y ( + ) y ( + ) ( ) (iii) y + 8 (b) Fin he following erivaive: (i) Page Topic C

14 C. Higher Orer Derivaive C. HIGHER ORDER DERIVATIVES The fir erivaive of a funcion i ielf a funcion an may be iffereniae. The erivaive of he fir erivaive of a funcion y f() i calle he econ erivaive an wrien a f "() [rea a "f ouble ah " ] Alernaive noaion are: y y (f()) [rea a "ee wo y by ee quare ] Noe: y i he fir erivaive quare. I i no he econ erivaive. The econ erivaive may be iffereniae again o obain he hir erivaive an o on o obain ucceively higher orer erivaive. EXAMPLE. Fin he fir an econ erivaive of he following funcion: (i) y + f() (i) y + f() y y f '() + 9 f "() 6 6 SAQ C. (a) Fin y given y + (b) Fin f "(), given f() + 6 Page Topic C

15 C.8 APPENDIX C.8 SOLUTIONS TO SELF-ASSESSMENT QUESTIONS SAQ C. Poiive an Negaive Power (a) f() f '() (b) f() f '() 6 (c) f() 0 f '() 0 9 () f() f '() (e) f() f '() 8 (f) f() f '() (g) f() f'() (h) f() 9 f'() SAQ C. Decimal an Fracional Power (a) f(). f'().. (b) f() 0.9 f'() (c) f(). f'().... () f() f'() (e) f() f'() (f) f() f'() (g) f() f'() (h) f() f'() SAQ C. Conan/ Conan Muliple of a funcion (a) f() f '() () 8 (b) f() f '() (c) f() 6 f '() (6 ) () f() (e) f() f '() ( ) f '() ( ) (f) f() f '() 6 6 Page Topic C

16 C.8 APPENDIX SAQ C. Sum (or Difference) of Funcion (a) f() + f '() + 6 (b) f() 6 8 f '() (c) f() f '() + 6 () (f) f() 6 f() f'() () ( + ) (e) f() f' () f'() SAQ C. Furher Eample (a) (i) y y y y ( ) + (iii) y y 0 (b) (i) (u u ( u + u ) u + + ) u Page 6 Topic C

17 C.8 APPENDIX SAQ C.6 Furher Eample (a) (i) y ( + ) y y ( + ) ( ) ( + + )( ) y (iii) y y (b) (i) + ( + ) ( + )( ) + + ( ) SAQ C. (a) y y y 0 Higher Orer Derivaive (b) f() f'() 6 f"() Page Topic C