lim Prime notation can either be directly applied to a function as previously seen with f x 4.1 Basic Techniques for Finding Derivatives

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1 MATH 040 Notes: Unit Page 4. Basic Techniques for Fining Derivatives In the previous unit we introuce the mathematical concept of the erivative: f f ( h) f ( ) lim h0 h (assuming the limit eists) In this unit we will look at rules for fining erivatives that will be simpler than applying the efinition. Prime notation can either be irectly applie to a function as previously seen with f or can be attache to the secon (epenent) variable of an equation such as y (assuming that y is a function of. Another common notation for erivatives is Leibniz notation, name in honor of the 7 th century co-iscoverer of calculus. The notation reinforces the concept of rate of change in a function by replacing the symbol elta with the letter. Leibniz notation may be written as f for an eplicit function f or as y when y is a function of. The notation may be use to inicate the intention to fin the erivative of an epression without eplicitly efining it as a function. On the following pages we will present a collection of rules for fining erivatives efficiently an effectively. It will be important to keep information organize an balance the work shown to arrive at a result: a minimum amount (for clarity of presentation an reaing) versus a sufficient amount (for clarity of mechanics an comprehension). 08

2 MATH 040 Notes: Unit Page Rules for Differentiation E : Fin the erivative. Rule : Constant Rule For any real constant k, a. f k 0. (If f = k, then f = 0.) b. 4 Rule : Power Rule For any constant eponent n, n (If f = n n. n, then f = n n.) c. f 7 E : Fin the erivative. E : Fin the erivative. a. a. b. f b. f 5 08

3 MATH 040 Notes: Unit Page E 4: Fin the erivative. a. f E 5: Fin the erivative. a. h b. g 4 5 b. v Rule : Constant Multiple Rule For any function f an a constant k, k f k f. (If f = kg, then f = kg.) E 6: Fin the erivative. 5 a. 0 b. f

4 MATH 040 Notes: Unit Page 4 Rule 4: Sum/Difference Rules For any functions f an g, f g f g. (If f = uv, then f = u v.) E 7: Fin the erivative. g 85 a. 4 Two notable shortcuts base on rules an : h 5 b. 6 an c c E 8: Fin the erivative. 5 f4 E 9: Fin the erivative. f 08

5 MATH 040 Notes: Unit Page 5 E 0: Fin the erivative. f 4 6 E : Fin the erivative. 5 f Marginal Analysis Given a revenue function R, recall that a marginal represents the revenue earne from selling the ( + ) st unit when units have alreay been sol, an there are analogous marginal for cost an profit functions, C an R. For linear functions, the marginal can be represente by the slope an will accurately represent the aitional unit. For nonlinear functions however, the closest approimation to the marginal comes from the concept most associate with slope of a function at a point: the instantaneous rate of change ientifies by the erivative. The erivative of a function can be use to fin an approimation of the revenue (or cost or profit) generate by selling one more unit. While the erivative won t necessarily prouce an eact etermination of the aitional revenue, it prouces an easy, meaningful approimation that in most cases is very close to the eact amount. 08

6 MATH 040 Notes: Unit Page 6 For a revenue function R, the notation for the marginal revenue function is an the calculation of marginal revenue is foun by (). Again, analogous relationships can be establishe between the cost function C an its marginal cost function an the profit function P an its marginal profit function. You may also see notation of the marginal functions as MR, MC, ormp, representing the marginal revenue, marginal cost, or marginal profit functions, respectively. Finally, some aitional reminers: If you wish to calculate the actual revenue generate by the ( + ) st unit, you can evaluate the revenue function at ( + ) an subtract it from the evaluation at. Given q for the quantity of an item an p for the price of an item, a eman function D can represent how one of these attributes affects the others. For instance, if we look at the price being a function of the eman in quantity, then p = D(q). Since revenue can be calculate by the prouct of q an p, then with this eman function, R(q) = D(q). E : A steel mill etermines that its cost function is Cn8 n6 n thousan ollars, where n is the amount of aily prouction of steel in tons. a. Calculate the cost of manufacturing 64 tons of steel per ay an approimate the cost of manufacturing 65 tons. Epress each in ollars. b. Fin the marginal cost function for C. c. Calculate an interpret in ollars the marginal cost of proucing one more ton of steel when 64 tons are being prouce each ay. 08

7 MATH 040 Notes: Unit Page 7 E : The eman function for heavyweight paper is 500 q p ollars, where q is 5 measure in reams. a. Fin the revenue function in terms of reams. c. Fin the marginal revenue function for R.. Calculate an interpret the marginal revenue prouce by selling one more ream when 00 reams are being sol. b. Calculate the revenue generate from 00 reams being sol. E 4: Continuing the previous eercise, suppose the cost function (in ollars) for heavyweight paper is given by Cq0 4 q, 0 q 00. c. Calculate an interpret the marginal profit function when 00 reams, 00 reams, an 50 reams are being prouce an sol. a. Fin the profit function P in terms of reams. b. Fin the marginal profit function for P. 08

8 MATH 040 Notes: Unit Page 8 4. The Chain Rule Recall from algebra that compose functions consist of one function insie a secon function. For eample, is consiere to be a composite function because the function ² eists within a cubing function (that is, the output of ² is the input to the cube). We can ecompose by efining the two functions that are brought together to make the new function. E: Decompose each of the following functions so that f(g()) returns the original function. a. 6 b. 4 f ( ) with g ( ) fg( ) c. 7 It is important to recognize composition functions because fining their erivatives requires a special rule calle the Chain Rule. E : Fin the erivative. a. y 6 Rule 5: Chain Rule f g ( ) f g ( ) g ( ) b. y 4 08

9 MATH 040 Notes: Unit Page 9 E : Fin the erivative. a. f 7 When Chain Rule is applie to a composite function whose outsie function f is a power function, as seen with the previous eamples, a simplifie variation of the rule can be applie calle Generalize Power Rule. Rule 5a: Generalize Power Rule b. f n g ng g n ( ) ( ) ( ) Applications of Chain Rule E : Suppose that for a group of 00,000 people, the number who survive to age is N Evaluate an interpret N an N = 6. E 4: For the first four years after purchase, the value of a $500 tablet computer can be estimate by function V t 500 t, 0t4. Determine the approimate value of the computer after two years an the rate of epreciation at that point in time. 08

10 MATH 040 Notes: Unit Page 0 4.4a/ 4.5a Basic Eponential & Logarithmic Derivatives Rule 6: Eponential Rule (if base is e) e e Rule 6*: Eponential Rule (if base is a) We can coming Chain Rule with the Eponential Rules to satisfy cases where the eponent of base (or a) is not simply. Rule 5a: e g e g g a g g ln a a g a ln a a Our primary but not eclusive focus will be on ifferentiating eponential epressions with a base of e. E : Fin the erivative. E : Fin the erivative. a. 4 y 0e a. f5e 6 b. f e 4 b. y e 6 8 c. f 5 c. f 4 08

11 MATH 040 Notes: Unit Page E : Base on Bureau of Labor Statistics ata, the eman for accountants & auitors.00t can be moele by Dt. e, where t is the number of years after 04 an D(t) is measure in millions. Calculate the number of accountants & auitors epecte to be neee in 00 an the rate of growth in eman. Epress values logically. E 4: A cup of coffee brewe at 90 egrees, if left in a 70 egree room, will cool to.04t T t 70 0e in t minutes. Calculate the temperature of the coffee in hour an the rate of change in the temperature at that time. Epress temperatures to the tenth of a egree. Rule 6: Logarithmic Rule (if base is e, > 0) ln Rule 6*: Logarithmic Rule (if base is a, > 0) log a lna Similar to eponentials, our primary but not eclusive focus will be on ifferentiating logarithmic epressions with a base of e. Recall that the omain of a logarithmic function is base on when the argument of the log is positive. As with eponentials, Chain Rule can be combine with Logarithmic Rules to cover cases where the argument is not simply. Rule 6a: g ln g, where g > 0 g g loga g ln( a) g 08

12 MATH 040 Notes: Unit Page E 5: Fin the erivative. a. yln8 E 6: Fin the erivative. a. fln b. f log4 b. ylog E 7: Fin the erivative. yln e E 8: Fin the erivative. yln 5 08

13 MATH 040 Notes: Unit Page E 9: The total revenue (in thousans of ollars) prouce by selling thousans of books can be epresse as R50ln4. The cost (in thousans of ollars) to prouce thousans of books is () =5. b. Fin the profit function an the marginal profit function. Interpret both when 0 thousan books are being sol. Roun to one ecimal place. a. Fin the marginal revenue function an interpret it when 0 thousan books are being sol. Roun to one ecimal place. E 0: Base on projections from, the typical resale value of a 00 Toyota Corolla sean can be projecte by ft logt with t equal to the number of years since 00 an f(t) in ollars. Calculate an interpret f an f t = 7. Epress in whole ollars. 08