SECTION 3.6 Derivatives of Logarithmic Functions 223 3.6 EXERCISES 1. Explain why the natural logarithmic functiony In x is used much more frequently in calculus than the other logarithmic functions y 33-34 Find an equation of the tangent line to the curve at the given point. log,x. 33. y In(x 3x +1), (3,0) 2-22 Differentiate the function. 34. y x2 In x, (1,0) 2. f(x)=x In x-x 3. f(x)= sin( In x) A35. If f(x) = sin x + In x, find f'(x). Check that your answer is reasonable by comparing the graphs of f andf'. 4. f(x)In(sinx) 5. f(x)= In 1 6. у X 36. Find equations of the tangent lines to the curve y = (In x)/x In x at the points (1,0) and (e, 1/e). Illustrate by graphing the curve and its tangent lines. 7. f(x)= log 10(1 +cos x) 8. f(x) log10 x 9. g(x) In(xe 2x) 37. Let f(x)= f'(T/4) 6? =cx +Incos x). For what value of c is 10. g(t) 1 +In t 11. F(t)=(In t) sin t 12. h(x) In(x + Vx2- 1) 3? 38. Let f(x) = log,(3x2 - 2 ) . For what value of b is f'(1) (2y1) Vy21 39-50 Use logarithmic differentiation to find the derivative of the function. In v 13. G(y) In 14. P(v) 1- e cosx 40. у 3 39. y (x2 2)(x4) 15. F(s) In ln s 16. y In 1+ t - t|| xe(x+ 1 17. T(z) 42. y 22 log2z cot x) 18. y n(csc x - 41. y x4 1 a2 z2 20. H(z)=In z2 44. y x 43. y x 19. y ln(e xe*) 46. y (x) 45. y xsinx log2 (x logs x) (sin x)n 21. y tan [In(ax + b)] 22. y 48. y 47. y (cos x)* 50. y (In x)osx 49. у %3 (tan x)/. 23-26 Find y' and y" In x 24. y V In x 51. Find y' if y In(x2 + y2 ). 23. у 3 1 + ln x 52. Find y' if x = y". 26. y In(1 + In x) 25. y In sec x 53. Find a formula for f(x) if f(x) = In(x - 1). d9 (x8 In x) dx 27-30 Differentiate f and find the domain of f. 54. Find 28. f(x) 2 + Inx X 27. f(x) 1 - ln(x 1) 55. Use the definition of derivative to prove that In(1+ x lim 30. f(x) In In In x 29. f(x) In(x2 2x) х = e* for any X 31. If f(x) In(x + In x), find f'(1). 56. Show that lim 1 п cos (In x2), find f'(1) 32. If f(x)

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