Example 1Example 2ExamplesPossible AnswersMAT 210 4.5 The Chain RuleExample 1Suppose that your hourly wage increases 40 cents per hour and your hours worked per week increasesby 5 hours. Then an expression for the change in your weekly gross income would be:(40centshour)(5hoursweek)=200centsweek Example 2Suppose that in a certain area, the level of CO2in the air is modeled by the equation C(p)=2 p,where the units of C(p) are parts per million (ppm) and p is the population. Also, suppose that thepopulation can be modeled by the equation p(t)=200 t2+3000, where t is the number of yearssince 1980. C is a function of p and p is a function of t. To find the instantaneous rate of change of Cwith respect to t, we will use the following rule.The Chain Rule (Form 1)dCdt=(dCdp)(dpdt)dCdp=2dpdt=400 tdCdt=(2ppmperson)(400 tpeopleyear)=800 tppmyearThe Chain Rule (Form 2)If f(x)=h(g(x)), then f'(x)=h'(g(x))⋅g'(x) In other words, multiply the derivative of the outside function by the derivative of the inside function.Examples1.y=(5 x−4)32. y=√3 x−2=(3 x−2)1/23. y=e3−2 xdydx=3(5 x−4)2⋅5¿15(5 x−4)2dydx=12(3 x−2)−1/2⋅3¿32(3 x −2)−1 /2dydx=−2 e3−2 x4.y=ln(4 x+5)5. y=32 x −4=3(2 x−4)−16. y=34 x −1dydx=14 x +5⋅4=44 x+5dydx=3(−1) (2 x−4)−2⋅2¿−6(2 x−4)−2dydx=34 x−1(ln 3)⋅4Exercises – Find the derivative of the functions in 1–91.y=(3 x2−4 x+1)32. y=13−2 x3. y=√3+4 x 4.y=ln(3 x−4)5. y=5 ex2−3 x6. y=23 x7.y=2 x+e1−3 x8. f(x)=(1−2ln x)39. y=(1−4 x− x3)510. Suppose that you invest $2000 in an account at 6% interest compounded monthly. (hint: Maple)a) Write an equation, B(t), for the balance after t years.b) Find an equation for the instantaneous rate of change of the balance after t years.c) Use it to find the rate at which the balance is changing 5 years after the account was opened.10. The percentage of households with TV’s who subscribed to cable from 1970 through 1990 can bemodeled by the equation P(t)=1001+14 . 96 e−0. 1527tpercentwhere t is the number of years since 1970.(hint: Maple)a) Write the rate of change formula for the percentage of households with TV’s who subscribe to cable.b) Use it to find how rapidly the percentage was growing in 1985.Possible Answers1.dydx=3(6 x−4)(3 x2−4 x+1)22. dydx=2(3−2 x)−23. dydx=2(3+4 x)−1/24.dydx=33 x−45. dydx=5(2 x−3)ex2−3 x6. dydx=23 x(ln 2)⋅37.dydx=2−3 e1−3 x8. f'(x)=−6(1−2 ln x)2x9.dydx=5(−4−3 x2)(1−4 x−x3)410. a) B(t)=2000(1+.0612)12 tb) B'(t)=2000(1. 005)12 t(ln1 . 005) (12)¿24000(ln 1. 005) (1. 005)12 tc)B'(5)=24000(ln1 .005) (1 . 005)60¿$161. 46 dollars/year10. a) P'(t)=−100(1+14 . 96 e−0 . 1527t)−2(14 . 96 e−0. 1527 t)(−0 .1527)b) P'(t)=−100(1+14 . 96e−0 . 1527(15))−2(1 4 . 96 e−0. 1527(15))(−0 . 1527)¿3 .66