DIFFERENTIAL CALCULUS  

One marks Questions  

1. If y = sin ( log x ) find dy / dx

2. Y = esin x

3. Y = e logx

x

4. Y = loge e(1 + sin h x )

5. Y = loge ( coshx)

6. If f ( x ) = 3 cos x + 4 sin x find f ` ( π/2)

7. If y = sin ( log x ). Then show that = 1 2

8. If y = 7x . x7 then find dy / dx =

9. If y = πx . xπ then find dy / dx.

10. Differentiate x w.r.t. logex.

11. Define the left hand derivative of a function y = f ( x ) at x = a

Two marks Questions:

12. If y = Tan-1 –

then find dy / dx

13. If y = log5 √ secx then find dy / dx

14. If y = e√x sinx find dy / dx

15. If y = log [ x + √ x2 + a2 ]. Prove that dy / dx, √

16. If y = ( x + √ x2 + 1 )5 then show that ( x2 + 1 ( dy / dx)2 = 25 y2

17. If y = log 1 1 cos prove that dy / dx = 2cosecx

18. If y = sin-1 then find dy / dx

19. If y = Tan-1 √1+x2 – 1 x

20. If y = Tan-1 3 3

1 3 2

21. If x5 y3 = ( x + 4 )8 then Prove that dy / dx. = y / x

22. If siny = x sin ( a + y ) prove that =

23. If y = sinhx + coshx then, find 1 / y dy / dx

24. If y = √ sin ( msin-1 √x ) then find dy / dx

25. If y = log [ x + √ x2 + a2 ] then show that dy / dx = √

26. If y = then Prove that dy / dx = - cosech2 x

27. If y = Tan-1

then find dy / dx

28. If y = Tan-1

then find dy / dx

29. If y = sec-1 1 2 1 2 then find dy / dx

30. If x2 + y2 = a2 then find dy / dx

Three marks Questions: 

31. If y = √ + sin-1 ( x / a) prove that dy / dx = 2 2

32. If y = Tan-1 √ √ √ √

then show that dy / dx = √

33. If √ 1 + 1 = a ( ( x – y ), prove that dy / dx

√

34. If x = a [ cost + loge Tan t / 2 ]., y = a sint. Then prove that dy / dx = Tan t.

35. Differentiate Tan-1

w,r,t cos-1 ( 2t2 – 1 )

36. If y = ( 1 + x )1/x + x (1 + x ) then find dy / dx 37. If y = x + ( sin-1 x )x then find dy / dx 38. If xm yn = ( x + y )m+n. then Prove that dy / dx = y / x 39. If xy = ey-x prove that dy / dx = 40. Differentiate Tan-1

√ w,r,t Sin 2

41. If x = a [ sin t – t cos t ] y = a [ cost + t sin t ] then prove that dy / dx at t = 3π/4 is “ – 1” 42. If y = Tan-1 √

√ then show that dy / dx =

√

43. If x = 3 cos t – 2 cos3t, y = 3 sint – 2 sin3 t. Prove that dy / dx = cot t.

44. If x = 3 sin2θ + 2 sin3 θ, y = 2 cos 3 θ – 3 cos2 θ prove that dy / dx = - Tan θ/2

45. Find the derivative of cos-1

w,r,t cot-1

46. If y = ( log x )x + ( sin-1 x)sinx then find dy / dx

47. If xy = yx prove that dy/dx =

48. If x2 + 2xy + 3y2 = 1 Prove that =

49. If x = a (θ - sin θ), y = a ( 1 – cos θ ) prove that = /

50. If y = x sinx , prove that x2 y2 – 2xy1 + ( x2 + 2 ) y = 0

51. If y = x sin h x . Prove that xy2 – 2y1 – xy + 2 sinhx = 0

52. If y = x cos ( log x ) + x logx, prove that x2 y2 – xy1 + 2y = x logx.

53. If y = ex logx. Prove that xy2 – ( 2x – 1 ) y1 + ( x – 1 ) y = 0

54. If y = Prove that ( 1 – x2 ) y2 – xy1 – m2y = 0

55. If y = ( cos-1 x)2 prove that ( 1 – x2) y2 – xy1 – 2 = 0

56. If y = eax sinbx. Prove that y2 – 2ay1 + ( a2 + b2 ) y = 0