Question 3: What are the derivatives of some basic functions (constant, linear, power, polynomial, exponential, and logarithmic)?

Transcription

1 Quesion : Wha are he erivaives of some basic funcions (consan, linear, power, polynomial, eponenial, an logarihmic)? There are many symbols use in calculus ha inicae erivaive or ha a erivaive mus be aken. The erivaive of y f( ) may be symbolize in any of he following ways: y f,, f( ) or D f For insance, he erivaive of he funcion f( ) 7 is f( ). Using he oher noaions for erivaive, we coul also wrie y 7 D 7 If he funcion uses a ifferen inepenen variable, we ajus he noaion o reflec his variable. The funcion has a erivaive g g () 7. This coul also be wrien as y 7 D 7 In many business applicaions he inepenen variable an epenen variables may have names o reflec wha hey represen. In a eman funcion, P DQ, he variable P represens he uni price of some iem an Q represens he number of iems sol a ha price. If he variables are relae by P Q, we can wrie he erivaive as 9

2 Q D Q 0.50Q P 0.50Q Q Q 0.50 Q D Q Q 0.50Q In his e, we will use each of hese noaions o help familiarize you wih each one. In mos non-mahemaics es, he auhor may have a favorie noaion an use ha noaion eclusively. The process of aking he erivaive of a funcion is calle iffereniaion. Aiionally, he erm iffereniae inicaes ha he erivaive shoul be aken. The erivaives of many funcions follow paerns. If we know hese paerns, we can avoi compuing erivaives from he efiniion irecly. The simples of hese paerns is he Consan Rule. The Consan Rule for Derivaives If f ( ) f ( ) is c, where c is a real number, hen he erivaive of f( ) 0 In simple erms, a consan funcion oes no change so he insananeous rae of change shoul be equal o zero. 0

3 Eample 5 Fin he Derivaive of a Consan Funcion Fin each of he erivaives below. a. F if F ( ) Soluion Since he erivaive of a consan is equal o zero, F 0 b. Soluion On he surface i migh appear ha he consan rule for erivaives oes no apply because of he power. Bu he erivaive is being aken wih respec o, no π. The symbol π is consan wih respec o so 0 Linear funcions are commonly encounere in business an economics. Since hey rise or fall a a consan rae, he erivaive of a linear funcion is simply he slope of he linear funcion. Derivaive of a Linear Funcion If f ( ) a b, where a an b are real numbers, hen he erivaive of f is f a

4 In his case, he slope of he linear funcion is a, he coefficien on he variable. Eample 6 Fin he Derivaive of a Linear Funcion Fin each of he erivaives below. a. y if y Soluion The erivaive of a linear funcion is equal o he coefficien on he variable, y b. D Soluion The variable in his linear funcion is, bu he rule for erivaives of linear funcions sill applies, D One of he mos useful erivaive formulas is he Power Rule for Derivaives. The Power Rule for Derivaives n If f ( ), where n is any real number, hen he erivaive of f is n f n

5 The Power Rule for Derivaives is easy o apply o funcions where he funcion has he form of a variable raise o a consan. In aiion, any funcion such as a roo funcion ha can be convere o a variable raise o consan can also be iffereniae wih he Power Rule for Derivaives. Eample 7 Fin he Derivaive of a Power Funcion Fin each of he erivaives below. a. Soluion Apply he Power Rule for Derivaives o give So. b. y if y.5 Soluion Alhough he variable in his funcion is no, we can sill apply he Power Rule in he form n n n. The Power Rule applies o power funcions wih any ype of number, such as a ecimal,in he power: y

6 The erivaive is c. r ( ) if r ( ) y Soluion Alhough his funcion oes no look like a power funcion, we can rewrie i as r ( ) an apply he Power Rule for Derivaives, r( ) Use an The erivaive is r( ). Eample 8 Fin he Derivaive of a Power Funcion Fin each of he erivaives below. a. 5 Soluion To be able o use he Power Rule for Derivaives, we nee o ake avanage of he fac ha 5 5. Apply he Power Rule o give 4

7 Use So 5 6 b. D ( Q) if DQ ( ) Q Soluion Rewrie he funcion using a negaive fracion eponen, Q Q, an apply he power rule, D( Q) Q Q Q Q 4 Q 4 Use Q 4 Q Q 4 4 The erivaive is D( Q). 4 Q Several rules mus be use o iffereniae polynomial funcions. The firs rule helps us o fin he erivaive of proucs where one of he facors is a consan. 5

8 Derivaives of a Consan Times a Funcion If a is a real number an hen f is a iffereniable funcion, af a f This rule ells us ha he erivaive of a consan imes a funcion is equal o he consan imes he erivaive of he funcion. Eample 9 Fin he Derivaive of a Power Funcion Muliplie by a Consan Fin each of he erivaives below. a. 5 0 Soluion Use he Prouc wih a Consan Rule an he Power Rule o calculae he erivaive, (0 ) Use he Consan Times a Funcion Rule Use he Power Rule 50 9 So

9 b. R ( Q) if RQ ( ) Q Soluion Use he Prouc wih a Consan Rule an he Power Rule o calculae he erivaive, R( Q) Q Q Q Q 0.765Q 0.5 Use he Consan Times a Funcion Rule Use he Power Rule 4.48Q 0.5 The erivaive is 0.5 R( Q) 4.48Q. The ne rule applies o iffereniaing sums or ifferences of funcions. Sum or Difference Rule for Derivaives If f ( ) an g ( ) are iffereniable funcions, hen f g f g The erivaive of a sum or ifference of funcions is equal o he sum or ifference of he erivaives of he funcions. This rule can be eene o ake he erivaive of any number of funcions ha are ae or subrace. If we combine his rule wih rule for aking erivaives of proucs wih a consan, we can ake he erivaive of any polynomial. 7

10 Eample 0 Fin he Derivaive of a Polynomial Fin each of he erivaives below. a. D.5 Soluion The erivaive of a sum or ifference is he sum or ifference of he erivaives, D.5 D D.5 D D.5D D.5 Use he Sum / Difference Rule wih he Consan Times a Funcion Rule Use he Power Rule 9 So b. D z z z e Soluion Break he erivaive of a polynomial ino he sum an ifference of he erivaives of he erms, z z e 4 z z e z z z z 5 4 z z e z z z 4 4 z z 5 0 Use he Sum / Difference Rule wih he Consan Times a Funcion Rule Use Power Rule on he firs wo erms Take noe of he hir erm in his polynomial. Even hough he hir erm looks like a power funcion, i is a consan since e is a consan. Therefore he erivaive of he hir erm is zero. Puing all of hese erms ogeher gives 8

11 z z z e z z As we saw in an earlier quesion, he erivaive of he eponenial funcion remember. e is easy o If f e, he he erivaive of f is f e In oher wors, he erivaive of e is are foun wih anoher simple formula. e. Eponenial funcions wih a base oher han e If f a, where a is a posiive real number, hen he erivaive of f is ln f a a Eample Fin he Derivaive of Eponenial Funcions Fin each of he erivaives. a. f ( ) if f ( ) 7.e Soluion Use he Prouc wih a Consan Rule o help ake he erivaive of he eponenial funcion, 9

12 f ( ) 7.e 7. e 7. e Use he Consan Times a Funcion Rule The erivaive is f ( ) 7.e b. S () if S ( ) 7,000 Soluion Use he Consan Rule o help ake he erivaive of he eponenial funcion, S( ) 7,000 7,000 7,000 ln Use Consan Times a Funcion Rule The erivaive is S( ) 7,000 ln. c. Soluion Since he power in he eponenial is insea of, we canno use he erivaive rule for eponenial funcions irecly. 9. In However, we can rewrie his eponenial funcion as his forma, we can apply he erivaive rule for eponenial funcions o yiel 9 ln 9 9 0

13 So ln99. The las ype of funcion we ll learn o iffereniae is a logarihmic funcion. If f log a erivaive of f, where a is a posiive real number, hen he is log a ln a In he case where he base of he logarihm is e, his relaionship simplifies consierably because: Since loge ln e log e is he same as ln, we can wrie anoher basic erivaive. If f ln, he erivaive of f is ln Eample Fin he Derivaive of Logarihmic Funcions Fin each of he erivaives. a. g ( ) if g( ) 5ln( )

14 Soluion For his funcion, we can use he Prouc wih a Consan Rule prior o aking he erivaive of he logarihm o give g( ) 5ln 5 ln 5 The erivaive is 5 g( ). b. D ln Soluion In his par, here is a insea of a in he logarihm as require by he erivaive rule for logarihms. However, he properies of logarihms allow he power in he logarihm o be move ousie of he logarihm, ln ln logarihms,. Now we can apply he erivaive rule for ln D ln D D ln Use he rules for simplifying b log blog logarihms, Use he Consan Times a Funcion Rule a a So D ln z c. log 9z Soluion Use he prouc propery of logarihms o wrie

15 z z log 9 log 9 log The erivaive can now be worke ou wih he erivaive rule for logarihms, log9z log9 logz z z log9 logz z z 0 ln() z Use Prouc Propery of Logarihms, y y log log log a a a Use Sum Rule for Derivaives In he firs erm, he erivaive of a consan is zero z So log 9z ln() z