Exercises and Problems for Section 3.6ExercisesFor Exercises 1-33, find the derivative. It may be to your ad- 21. h(w) = warcsin w22. f () = ein(kz)vantage to simplify before differentiating. Assume a, b, c, and& are constants.23. r(t) = arcsin(2t)24. j(x) = cos (sin 1 x)1. f (t) = In(12 + 1)2. f(:) = In(1 – x)25. f() = cos(arctan 3x) 26. f(2) =3. f(x) = 1(ex)In4. f(z ) = eln(212+3)5. f(x) = In(1 – e-*)6. f(a) = In(sina)27. f(x) =1+ Inc7. f(x) = In(e# + 1)8. y= xle- I+ 29. j(I) = In(caz + 0)10. h(w) = w3 In(10w)28. y = 2x(Inx + In 2) – 2x +e11. f(x) = In(e7)12. f(2) = e(lux)+129. f(x) = In(sin x + cos z)13. f(w) = In(cos(w -1))14 . f (t ) = In(eint )30. f(t) = In(Int) + In(In 2)15. f(3/) = arcsin(y?)16. g(t) = arctan(3t – 4) 31. T(u) = arctan ( 14 )17. g(a) = sin(arcsin a)18. g(t) = carctan(31? )32. a(t) = In (_-cost(1 + cost/19. g(t) = cos(In t)20. h(z) = 2 233. f() – costarcsin(I + 1))ProblemsFor Problems 34-37, let h(x) = f(g(x)) and k(x) = 40. Using the chain rule, find -1 (log x).o(f(I)). Use Figure 3.29 to estimate the derivatives.(Recall log z. = log 10 .)41. To compare the acidity of different solutions, chemists13use the pH (which is a single number, not the product off(2)g(x)p and H). The pl is defined in terms of the concentra-tion, , of hydrogen ions in the solution as-3pll = – log :c.Find the rate of change of pH with respect to hydrogenFigure 3.29ion concentration when the pH is 2. [Hint: Use the resultof Problem 40.]42. A firm estimates that the total revenue, R, in dollars, re-34. h'(1)35. k'(1) 36. H'(2)37. k'(2)ceived from the sale of q goods is given byR = In(1 + 1000q?).38. On what intervals is In (c’ + 1) concave up?39. Use the chain rule to obtain the formula forThe marginal revenue, MR, is the rate of change ofthe total revenue as a function of quantity. Calculate thedo (arcsin x).marginal revenue when q = 10.
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