DIFFERENTIAL CALCULUS One marks Questions 1. If y = sin ( log x ) find dy / dx
2. Y
3. Y =
= esin x
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
18. If y = sin-1
1
1 cos
prove that dy / dx = 2cosecx
then find dy / dx
√1+x2 – 1
19. If y = Tan-1
x
-1
20. If y = Tan
3
3
2
1 3
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
√
then Prove that dy / dx = - cosech2 x
y =
27. If y = Tan-1
then find dy / dx
28. If y = Tan-1
then find dy / dx
29. If y = sec-1
1
1
2
then find dy / dx
2
30. If x2 + y2 = a2 then find dy / dx
Three marks Questions: 31. If y =
√
32. If y = Tan-1
33. If √ 1
sin-1 ( x / a) prove that dy / dx =
+
+
√
√
√
√
1
then show that dy / dx
= a ( ( x – y ), prove that dy / dx
=
2
2
√
√
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
x + ( sin-1 x )x then find dy / dx
37. If y =
38. If xm
yn = ( x + y )m+n. then Prove that dy / dx = y / x
39. If xy = ey-x prove that dy / dx =
w,r,t Sin
40. Differentiate Tan-1
√
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