Derivative rules for composite and implicit functions - Question Bank

1. If y = e^(cos(x)), find dy/dx.
A) -e^(cos(x)) sin(x)
B) e^(cos(x)) sin(x)
C) -sin(x) e^(sin(x))
D) cos(x) e^(cos(x))
2. What is the derivative of f(x) = sin^{-1}(x^2)?
A) 2x / sqrt(1 - x^4)
B) 1 / sqrt(1 - x^4)
C) 2x cos(x^2)
D) x^2 / sqrt(1 - x^4)
3. If y = tan(x^2 + x), find dy/dx.
A) sec^2(x^2 + x) * (2x + 1)
B) sec^2(x^2 + x)
C) (2x + 1) tan(x^2 + x)
D) tan(2x + 1)
4. Find the derivative of y = ln(x^2 + 1).
A) 2x / (x^2 + 1)
B) 1 / (x^2 + 1)
C) 2x ln(x^2 + 1)
D) 1 / x
5. If y = x^2 * e^(sin(x)), find dy/dx.
A) 2x e^(sin(x)) + x^2 e^(sin(x)) cos(x)
B) 2x e^(sin(x))
C) x^2 e^(sin(x)) cos(x)
D) e^(sin(x)) + x^2 e^(sin(x)) cos(x)
6. What is the derivative of f(x) = cos(e^x)?
A) -e^x sin(e^x)
B) -sin(e^x)
C) e^x cos(e^x)
D) -e^x cos(e^x)
7. Find the derivative of y = x * sin^{-1}(x).
A) sin^{-1}(x) + x / sqrt(1 - x^2)
B) 1 / sqrt(1 - x^2)
C) x / sqrt(1 - x^2)
D) sin^{-1}(x)
8. If y = sqrt(x^2 + a^2), find dy/dx.
A) x / sqrt(x^2 + a^2)
B) a / sqrt(x^2 + a^2)
C) x / (2 * sqrt(x^2 + a^2))
D) 1 / (2 * sqrt(x^2 + a^2))
9. What is the derivative of f(x) = tan(e^x)?
A) sec^2(e^x) * e^x
B) sec^2(e^x)
C) e^x tan(e^x)
D) tan(e^x)
10. Find the derivative of y = (sin(x))^n.
A) n (sin(x))^(n-1) cos(x)
B) n (sin(x))^(n-1)
C) n cos(x) (sin(x))^(n-1)
D) (sin(x))^n cos(x)
11. If y = x^2 * ln(x), find dy/dx.
A) 2x ln(x) + x
B) x ln(x) + x^2
C) 2x ln(x)
D) x + ln(x)
12. What is the derivative of f(x) = ln(tan(x))?
A) sec(x) csc(x)
B) sec(x) / tan(x)
C) csc(x) / tan(x)
D) sec^2(x)
13. Find the derivative of y = e^(x^3).
A) 3x^2 e^(x^3)
B) x^2 e^(x^3)
C) e^(x^3)
D) 3x^2
14. If y = sin(x^2) + cos(x^2), find dy/dx.
A) 2x cos(x^2) - 2x sin(x^2)
B) cos(x^2) - sin(x^2)
C) 2x cos(x^2) + 2x sin(x^2)
D) sin(x^2) - cos(x^2)
15. What is the derivative of f(x) = (x^2 + 1)^3?
A) 3(x^2 + 1)^2 * 2x
B) (x^2 + 1)^2 * 2x
C) 3(x^2 + 1)^2
D) 6x(x^2 + 1)^2
16. Find the derivative of y = x^2 * cos(x).
A) 2x cos(x) - x^2 sin(x)
B) 2x cos(x) + x^2 sin(x)
C) -x^2 sin(x)
D) 2x cos(x)
17. If y = e^(tan^{-1}(x)), find dy/dx.
A) e^(tan^{-1}(x)) / (1 + x^2)
B) e^(tan^{-1}(x))
C) 1 / (1 + x^2)
D) tan^{-1}(x) e^(tan^{-1}(x)-1)
18. What is the derivative of f(x) = arcsin(x/a)?
A) 1 / sqrt(a^2 - x^2)
B) a / sqrt(a^2 - x^2)
C) 1 / sqrt(1 - x^2)
D) 1 / (a^2 + x^2)
19. Find the derivative of y = ln(sin(x)).
A) cot(x)
B) tan(x)
C) -cot(x)
D) sec(x)
20. If y = x^3 * e^(2x), find dy/dx.
A) 3x^2 e^(2x) + 2x^3 e^(2x)
B) 3x^2 e^(2x)
C) 2x^3 e^(2x)
D) x^3 e^(2x) + 2x^2 e^(2x)
21. What is the derivative of f(x) = sqrt(sin(x))?
A) cos(x) / (2 * sqrt(sin(x)))
B) sin(x) / (2 * sqrt(cos(x)))
C) sqrt(cos(x))
D) 1 / (2 * sqrt(sin(x)))
22. If y = cos(x^2 + 1), find dy/dx.
A) -2x sin(x^2 + 1)
B) 2x sin(x^2 + 1)
C) -sin(x^2 + 1)
D) -2x cos(x^2 + 1)
23. Find the derivative of y = e^(x^2 + x).
A) (2x + 1) e^(x^2 + x)
B) x e^(x^2 + x)
C) 2x e^(x^2 + x)
D) e^(2x + 1)
24. If y = x^n, find dy/dx.
A) nx^(n-1)
B) n x^n
C) x^(n-1)
D) n x
25. What is the derivative of f(x) = tan^{-1}(x^2)?
A) 2x / (1 + x^4)
B) 1 / (1 + x^4)
C) 2x sec^2(x^2)
D) x^2 / (1 + x^4)
26. If y = x * arctan(x), find dy/dx.
A) arctan(x) + x / (1 + x^2)
B) 1 / (1 + x^2)
C) arctan(x)
D) x / (1 + x^2)
27. Find the derivative of y = sin^{-1}(e^x).
A) e^x / sqrt(1 - e^(2x))
B) 1 / sqrt(1 - e^(2x))
C) e^x / sqrt(1 - e^x)
D) e^x cos(e^x)
28. If y = (ln(x))^2, find dy/dx.
A) 2 ln(x) / x
B) ln(x) / x
C) 2 ln(x)
D) 1 / x
29. What is the derivative of f(x) = log(1 + x^2) with respect to x?
A) 2x / (1 + x^2)
B) 1 / (1 + x^2)
C) 2x log(1 + x^2)
D) log(2x)
30. Find the derivative of y = x^2 * sin(x).
A) 2x sin(x) + x^2 cos(x)
B) 2x cos(x) + x^2 sin(x)
C) x^2 cos(x)
D) 2x sin(x)
31. If y = e^(sin(x)), find dy/dx.
A) e^(sin(x)) cos(x)
B) e^(sin(x))
C) cos(x) e^(cos(x))
D) sin(x) e^(sin(x))
32. What is the derivative of f(x) = cot(x^2)?
A) -2x csc^2(x^2)
B) csc^2(x^2)
C) -csc^2(x^2)
D) 2x cot(x^2)
33. Find the derivative of y = sin(ax + b).
A) a cos(ax + b)
B) cos(ax + b)
C) -a sin(ax + b)
D) a sin(ax + b)
34. If x^3 + y^3 = 3xy, find dy/dx.
A) (y - x^2) / (y^2 - x)
B) (x^2 - y) / (y^2 - x)
C) (3y - 3x^2) / (3y^2 - 3x)
D) (y^2 - x) / (x^2 - y)
35. What is the derivative of f(x) = sec(x^3)?
A) 3x^2 sec(x^3) tan(x^3)
B) sec(x^3) tan(x^3)
C) 3x^2 sec(x^3)
D) sec(3x^2) tan(3x^2)
36. If y = sqrt(ax + b), find dy/dx.
A) a / (2 * sqrt(ax + b))
B) 1 / (2 * sqrt(ax + b))
C) a * sqrt(ax + b)
D) b / (2 * sqrt(ax + b))
37. Find the derivative of y = x^n * e^x.
A) nx^(n-1)e^x + x^n e^x
B) x^n e^x + x^(n-1) e^x
C) n x^n e^x
D) x^n e^x
38. If y = log_a(x), find dy/dx.
A) 1 / (x ln(a))
B) 1 / x
C) ln(a) / x
D) 1 / (x log_a(e))
39. Given y = e^(kx), find dy/dx.
A) k e^(kx)
B) e^(kx)
C) k e^(x)
D) e^(kx) / k
40. What is the derivative of f(x) = arcsin(2x)?
A) 2 / sqrt(1 - 4x^2)
B) 1 / sqrt(1 - 4x^2)
C) 2 / sqrt(1 - x^2)
D) 1 / sqrt(1 - x^2)
41. If y = tan(3x), what is dy/dx?
A) sec^2(3x)
B) 3 sec^2(3x)
C) tan(3x)
D) 3 tan(3x)
42. Find the derivative of y = sin(x)cos(x).
A) cos^2(x) - sin^2(x)
B) 2cos^2(x) - 1
C) 1 - 2sin^2(x)
D) cos(2x)
43. If x^2 + y^2 = 25, find dy/dx.
A) -x/y
B) x/y
C) -y/x
D) y/x
44. What is the derivative of y = arctan(x/a)?
A) 1 / (1 + (x/a)^2)
B) a / (a^2 + x^2)
C) 1 / (a^2 + x^2)
D) 1 / sqrt(1 - (x/a)^2)
45. Differentiate f(x) = a^(g(x)) where g(x) is a function of x.
A) a^(g(x)) g'(x) ln(a)
B) a^(g(x)) g'(x)
C) g'(x) a^(g(x)-1)
D) a^(g'(x)) ln(a)
46. If y = x^x, what is dy/dx?
A) x^x (1 + ln(x))
B) x^x (ln(x))
C) x * x^(x-1)
D) x^x
47. Find the derivative of h(x) = cos(sin(x)).
A) -sin(sin(x)) cos(x)
B) sin(sin(x)) cos(x)
C) -cos(cos(x)) sin(x)
D) sin(cos(x)) sin(x)
48. If y = sqrt(1 - x^2), find dy/dx.
A) 1 / (2 * sqrt(1 - x^2))
B) -x / sqrt(1 - x^2)
C) x / sqrt(1 - x^2)
D) -1 / sqrt(1 - x^2)
49. What is the derivative of g(x) = ln(cos(x))?
A) tan(x)
B) -tan(x)
C) cot(x)
D) -cot(x)
50. Given f(x) = e^(tan(x)), what is f'(x)?
A) e^(tan(x)) sec^2(x)
B) e^(tan(x)) tan(x)
C) sec^2(x) e^(tan(x)-1)
D) e^(sec^2(x))
All Question Banks
E-Book Derivative rules for composite and implicit functions One Line Questions Derivative rules for composite and implicit functions Online Test Derivative rules for composite and implicit functions