d In ( x 4 + 1 ) = - 1 C In Problems 57-62, find f (x ) and find the eq 15. dx x 4 + 1 tangent to the graph of f at the indicated value 57. f (x ) = x(4 - x) '; x = 2 16. d In ( x - x3 ) = dx 58. f ( x) = x2 ( 1 - x) 4; * = 2 In Problems 17-34, find f' (x) and simplify. 59 . f (x ) = (2x - 5) 3; * = 3 17 . f ( x ) = (5 - 2x ) 4 18. f (x ) = (9 - 5x) 2 19 . f ( x ) = ( 4 + 0 . 2 * ) 5 20 . f (x ) = (6 - 0.5x) 4 60. f(x) = (3x - 8) 2' *= 4 21. f (x ) = ( 3x2 + 5 ) 5 22. f ( x ) = ( 5x2 - 3) 6 61. f( x ) = Vinx; x = e 23 . f (x ) = esx 24. f(x ) = 6e -2x 62. f(x) = evr; x =1 25. f (x ) = 3e-6x 26. f(x ) = @12+ 3x + 1 27 . f (x ) = (2x - 5) 1/2 . In Problems 63-68, find f' (x) and find the 28. f ( x ) = ( 4x + 3 ) 1/2 tangent line is horizontal. 29 . f (x ) = ( x 4 + 1 ) - 2 30. f(x) = (x5 + 2 )-3 63. f(x) = x2(x - 5)3 64. f(x) 31. f(x) = 3 In(1 + x2) 32. f(x) = 2In(x2 - 3x +4) 65. f (x ) = 33. f ( x ) = (1 + Inx) 3 34. f ( x ) = (x - 2 Inx)4 ( 2x + 5 ) 2 66. f(x In Problems 35-40, find f' (x) and the equation of the line 67. f(x) = Vx2 - 8x + 20 68. f(x tangent to the graph off at the indicated value of x. Find the 69. A student reasons that the functions f value(s) of x where the tangent line is horizontal. and g (x) = 4 In(x2 + 3) must have 35. f(x) = ( 2x - 1)3; x = 1 since he has entered f(x ), 8 (x), f' (x 36. f(x) = (3x - 1 ) 4; x = 1 graphing calculator, but only three gra figure). Is his reasoning correct? Are 37. f ( x) = (4x - 3) 1/2; x = 3 same function? Explain. 38. f (x) = (2x + 8) 1/2; x = 42 NORMAL FLOAT AUTO REAL DEGREE MP NORMAL FLO 39. f (x ) = 5et - 4x+ 1; x = 0 Plotz 40. f ( x) = In( 1 - x2 + 2x4 ) ; x = 1 \\Y1Bin(5(X2+3)") Y 284In(X2+3) 2 + er ) 4 B In Problems 41-56, find the indicated derivative and simplify. \\Y 3 B dx (Y1) |x=x 41. y' if y = 3(x2 -2)4 42. y' if y =2(x3+ 6)s "\\Y 4 2x (Yz) | x=x KY 5= INY 6= OK 43 . - 2 ( 12 + 31 ) - 3 dt dt 44. " 3 (1 3 + 1 2 ) - 2(14 dh 45. 7w if h( w ) = Vw-+ 8ta son (A) Figure for 73 dg 46. dw if 8 (w ) = V3w - Jar goodenone 70. A student reasons that the function on that will make the f (x ) = (x+ 1)In(x+ 1) - 47. 8' ( x) if 8(x) = 4xe3x must have the same derivative sin 8 (x), f' (x), and g' (x) into a gr three graphs appear (see the figur 48. h' (x ) if h (x ) = Are f' (x) and g' (x) the same f 49. d In ( 1 + x 2 ) . ~ [x In ( 1 + er ) ] NORMAL FLOAT AUTO REAL DEGREE MP NORMAL dx 3 x Ploti Plot2 Plot3 51 . F' ( t ) if F ( t ) = (ef+1)3 natdog would \\Y18(X+1)In(X+1)-x Y 2 (X+1) 173 52. G' (t ) if G (t) = (1 - e2 ) 2 Y 3 dx (Y1) |x=x 53 . y' if y = In(x2 + 3) 3/2un botomicmartini \\Y 48 2x (Y2) |x=x NY 5= INY 6= 54. y' if y = [In(x2 + 3)13/2 INY 7= d d (A) = etr-? ? 55. 1 56. dw ( w' + 4 )s " dw ( w- - 2) 6