1. f'(x) = 12x² - 20x³
2. f'(x) = e^x sin x + e^x cos x
3. f'(x) = -4( x⁴ + 3x )⁻⁵ (4x³ + 3)
4. f'(x) = 6x(x²+1)⁶ + 6x³(x²+1)⁵
5. f'(x) = -sin x - 4x
6. f'(x) = (2x(x²-7) - x²(2x)) / (x²-7)²
7. f'(x) = (2x·x² - (x²-1)·2x) / x⁴
8. f'(x) = 3x² · 3x² + 3x³ · 6x
9. f'(x) = 1/(ln(x²)) · (2x)/x²
10. f'(x) = [ (6x²·x² - (2x³+3x²-1)·2x) ] / x⁴
11. f'(x) = 2x√(x²-2) + x²·(x/√(x²-2))
12. f'(x) = 2 - (4/3)x^(-1/3)
13. f'(x) = [ 4·3(3x-1)²·3·(x²+7)² - 4(3x-1)³·2(x²+7)·2x ] / (x²+7)⁴
14. f'(x) = -x / √(x²+8)
15. f'(x) = [ 1·√(1-(ln x)²) - x·(1/2)(1-(ln x)²)^(-1/2)·(-2 ln x · 1/x) ] / (1-(ln x)²)
16. f'(x) = [ 0·(3x²+x)³ - 6·3(3x²+x)²·(6x+1) ] / (3x²+x)⁶
17. f'(x) = [ 3·(3x²-π)²·6x ] / 6
18. f'(x) = [ 1·(x²+3x)⁵ - x·5(x²+3x)⁴·(2x+3) ] / (x²+3x)¹⁰
19. f'(x) = -5(e^x)⁻⁶ · e^x
20. f'(x) = 10[arctan(2x)]⁹ · (2/(1+4x²))
21. f'(x) = [ 1·(e^(2x)+e) - x·2e^(2x) ] / (e^(2x)+e)²
22. f'(x) = 4(x⁸+7)³·8x⁷(4x+7)³ + (x⁸+7)⁴·3(4x+7)²·4
23. f'(x) = 5(7x+√(x²+3))⁴ · (7 + x/√(x²+3))
24. f'(x) = [ (1/2)x^(-1/2)·(x-1) - (√x + 2/3)·1 ] / (x-1)²
25. f'(x) = (1/3)x^(-2/3) + (3/2)x^(-5/2)
26. f'(x) = [ (2/7)·(7x-9) - (2x/7+5)·7 ] / (7x-9)²
27. f'(x) = [ cos x · cos x - sin x · (-sin x) ] / cos²x
28. f'(x) = e^x(x²+3)(x³+4) + e^x(2x)(x³+4) + e^x(x²+3)(3x²)
29. f'(x) = [ (5x²-7x)·2x - (x²+2)·(10x-7) ] / (x²+2)²
30. f'(x) = 3[ln(5x²+9)]² · (10x)/(5x²+9)
31. f'(x) = (10x)/(5x²+9)
32. f'(x) = -csc²(6x) · 6
33. f'(x) = sec²x - sec²x
34. f'(x) = (1/√(1-4^x)) · ln2 · 2^x
35. f'(x) = sec²(cos x) · (-sin x)
36. f'(x) = 2[(x²-1)² - x²] · [2(x²-1)·2x - 2x]
37. f'(x) = sec x tan x · sin(3x) + sec x · 3cos(3x)
38. f'(x) = [ 2(x-1)·(x+1)² - (x-1)²·2(x+1) ] / (x+1)⁴
39. f'(x) = [ 1/(3x²+4x) ] · (6x+4)
40. dy/dx = e^(2y) + 2x e^(2y) dy/dx
41. y + x dy/dx + 2y dy/dx = 0
42. (cos y dy/dx)(y²+1) - sin y (2y dy/dx) = 3
43. (g - f)'(2) = g'(2) - f'(2) = 2 - 3 = -1
44. (fg)'(2) = f'(2)g(2) + f(2)g'(2) = (3)(-5) + (7)(2) = -15 + 14 = -1
45. (f/g)'(2) = [ f'(2)g(2) - f(2)g'(2) ] / [g(2)]² = [ (3)(-5) - (7)(2) ] / (-5)² = [-15 - 14]/25 = -29/25
46. (5f + 3g)'(2) = 5f'(2) + 3g'(2) = 5(3) + 3(2) = 15 + 6 = 21
47. (f ∘ f)'(2) = f'(f(2)) · f'(2) = f'(7) · 3 = (unknown) · 3 → Cannot be determined with given info.
48. (f/(f+g))'(2) = [ f'(2)(f(2)+g(2)) - f(2)(f'(2)+g'(2)) ] / (f(2)+g(2))² = [ 3(7-5) - 7(3+2) ] / (7-5)² = [ 3(2) - 7(5) ] / 4 = [6 - 35]/4 = -29/4
Parent Tip: Review the logic above to help your child master the concept of derivatives practice worksheet.
