MATH 171 - Derivative Worksheet Differentiate these for fun, or practice, whichever you need. The given answers are not simplified. 1. f (x) = 4x5 − 5x4 2. f (x) = ex sin x 3. f (x) = (x4 + 3x)−1 4. f (x) = 3x2 (x3 + 1)7 5. f (x) = cos4 x − 2x2 6. f (x) = 7. f (x) = 10. f (x) = x2 − 1 x 1 2x4 + 3x2 − 1 x2 8. f (x) = (3x2 )(x 2 ) 9. f (x) = ln(xe7x ) √ 11. f (x) = (x3 ) 5 2 − x 4 12. f (x) = 2x − √ x 4(3x − 1)2 13. f (x) = x2 + 7x 14. f (x) = √ 6 2 (3x − π)4 17. f (x) = (3x2 − πx)4 6 16. f (x) = 6 x2 + 8 15. f (x) = q 18. f (x) = i10 h 19. f (x) = (xex )π 3 22. f (x) = (x + 1) (4x + 7) 23. f (x) = (7x + √ x2 x 1 − (ln x)2 (x2 + x √ 3x)5 1 21. f (x) = (e2x + e) 2 20. f (x) = arctan(2x) 5 x 1 + x2 6 + 3) 24. f (x) = 1 x + 1 x2 x−1 26. f (x) = s 28. f (x) = ex (x2 + 3)(x3 + 4) 29. f (x) = 5x2 − 7x x2 + 2 31. f (x) = ln(5x2 + 9)3 32. f (x) = cot(6x) 33. f (x) = sec2 x · tan x 34. f (x) = arcsin(2x ) 35. f (x) = tan(cos x) 36. f (x) = [(x2 − 1)5 − x]3 37. f (x) = sec x · sin(3x) 38. f (x) = 25. f (x) = √ 3 1 x2 − √ x3 In problems 40 – 42, find 40. 3y = xe5y 2x + 5 7x − 9 (x − 1)3 x(x + 3)4 27. f (x) = sin x cos x h 30. f (x) = ln(5x2 + 9)]3 39. f (x) = log5 (3x2 + 4x) dy . Assume y is a differentiable function of x. dx 41. xy + y 2 + x3 = 7 42. sin y = 3x y2 + 1 If f and g are differentiable functions such that f (2) = 3 , f ′ (2) = −1 , f ′ (3) = 7 , g(2) = −5 and g ′ (2) = 2 , find the numbers indicated in problems 43 – 48. 43. (g − f )′ (2) ′ 46. (5f + 3g) (2) 44. (f g)′ (2) ′ 47. (f ◦ f ) (2) !′ 45. f g 48. f f +g (2) !′ (2) Answers: Absolutely not simplified ... you should simplify more. 1. f ′ (x) = 20x4 − 20x3 2. f ′ (x) = ex cos x + (sin x)ex 3. f ′ (x) = −1(x4 + 3x)−2 (4x3 + 3) 4. f ′ (x) = 3x2 · 7(x3 + 1)6 (3x2 ) + (x3 + 1)7 · 6x 5. f ′ (x) = 4(cos x)3 (− sin x) − 4x 6. f ′ (x) = (1 + x2 )(1) − x(2x) (1 + x2 )2 5 3 8. f ′ (x) = 3 · x 2 (Simplify f first.) 2 7. f ′ (x) = 1 + x−2 (Simplify f first.) 1 + 7 (Simplify f first.) 10. f ′ (x) = 4x + 0 + 2x−3 (Simplify f first.) x −3 −4 1 1 12. f ′ (x) = 2 + 2x 2 11. f ′ (x) = x3 · (2 − x) 5 (−1) + (2 − x) 5 (3x2 ) 5 9. f ′ (x) = h i 13. f ′ (x) = (x2 + 7x ) 4 · 2(3x − 1)(3) − 4(3x − 1)2 (2x + 7x ln 7) 15. f ′ (x) =  17. f ′ (x) = i 1h 4(3x2 − πx)3 (6x − π) 6 (x2 1 − (ln x)2 1 2 (1) − x · 1 2 +  1 − (ln x)2 h i  −1  2 1 x − 2(ln x) ·  16. f ′ (x) = −24(3x2 − π)−5 (6x)  h i √ √ −1 (x2 + 3x)5 (1) − x 5(x2 + 3x)4 2x + 12 (3x) 2 · 3 √ 18. f ′ (x) = (x2 + 3x)10 h i9 1 20. f ′ (x) = 10 arctan(2x) · ·2 1 + (2x)2 1 − (ln x)2 19. f ′ (x) = π(xex )(π−1) xex + ex −1 1 14. f ′ (x) = (x2 + 8) 2 (2x) 2 7x )2 h h i i −1 1 21. f ′ (x) = (e2x + e) 2 (e2x · 2 + 0) 22. f ′ (x) = (x6 + 1)5 3(4x + 7)2 (4) + (4x + 7)3 5(x6 + 1)4 (6x5 ) 2   √ −1 1 (x − 1)(−x−2 − 2x−3 ) − (x−1 + x−2 )(1) 23. f ′ (x) = 6(7x + x2 + 3)5 7 + (x2 + 3) 2 · 2x 24. f ′ (x) = 2 (x − 1)2 1 2x + 5 26. f ′ (x) = 2 7x − 9 2 −1 3 −5 25. f ′ (x) = x 3 + x 2 3 2 27. f ′ (x) = sec2 x h i (x2 + 2)(10x − 7) − (5x2 − 7x)(2x) (x2 + 2)2 h i 1 2 2 31. f ′ (x) = · 3(5x + 9) (10x + 0) (5x2 + 9)3   h 34. f ′ (x) = q h  i h · 1 (10x + 0) 5x2 + 9 h 1 1 − (2x )2 36. f ′ (x) = 3 (x2 − 1)5 − x 37. f ′ (x) = sec x cos(3x) · 3 + sin(3x) sec x tan x  i2   5(x2 − 1)4 · 2x − 1 i x2 (x + 3)8 1 · (6x + 4) 2 (3x + 4x) · ln 5 −3x2 − y dy = 41. dx x + 2y 44. 11 45. dy e5y = dx 3 − 5xe5y dy 3(y 2 + 1)2 42. = 2 dx (y + 1)(cos y) − 2y sin y 40. −1 25 i · 2x ln 2 x(x + 3)4 3(x − 1)2 (1) − (x − 1)3 x · 4(x + 3)3 (1) + (x + 3)4 (1) 39. f ′ (x) = 43. 3 (7x − 9)(2) − (2x + 5)(7) (7x − 9)2 32. f ′ (x) = − csc2 (6x) · 6 i 35. f ′ (x) = sec2 (cos x) (− sin x) i2 h h 38. f ′ (x) = 2 30. f ′ (x) = 3 ln(5x2 + 9) 33. f ′ (x) = sec2 x(sec2 x) + tan x 2 · sec x(sec x tan x)   −1 " 28. f ′ (x) = ex (x2 + 3) (3x2 ) + (x3 + 4) ex (2x) + (x2 + 3)ex 29. f ′ (x) =   46. 1 47. −7 48. −1 4 #
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