FAC1004: Rapid-Fire Drill Pack

Objective: Maximize problem volume in minimal time. Pure pattern recognition and mechanical fluency.
Target: 60–90 seconds per problem. If you stall >3 minutes, skip and mark it.
Total problems: 72
Estimated time: 2–3 hours (or split into three 45-minute sprints)

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Cheat Sheet (Memorize First)

Inverse Trig Derivatives

Function	Derivative
$\sin^{-1} u$	$\frac{u'}{\sqrt{1-u^2}}$
$\cos^{-1} u$	$-\frac{u'}{\sqrt{1-u^2}}$
$\tan^{-1} u$	$\frac{u'}{1+u^2}$
$\cot^{-1} u$	$-\frac{u'}{1+u^2}$
$\sec^{-1} u$	$\frac{u'}{u\sqrt{u^2-1}}$
$\csc^{-1} u$	$-\frac{u'}{u\sqrt{u^2-1}}$

Hyperbolic Definitions

$$\sinh x = \frac{e^x-e^{-x}}{2}, \quad \cosh x = \frac{e^x+e^{-x}}{2}, \quad \tanh x = \frac{\sinh x}{\cosh x}$$ $$\cosh^2 x - \sinh^2 x = 1, \quad 1 - \tanh^2 x = \text{sech}^2 x, \quad \coth^2 x - 1 = \text{csch}^2 x$$

Hyperbolic Derivatives

Function	Derivative
$\sinh u$	$\cosh u \cdot u'$
$\cosh u$	$\sinh u \cdot u'$
$\tanh u$	$\text{sech}^2 u \cdot u'$
$\coth u$	$-\text{csch}^2 u \cdot u'$
$\text{sech } u$	$-\text{sech } u \tanh u \cdot u'$
$\text{csch } u$	$-\text{csch } u \coth u \cdot u'$

Inverse Hyperbolic Derivatives

Function	Derivative
$\sinh^{-1} u$	$\frac{u'}{\sqrt{1+u^2}}$
$\cosh^{-1} u$	$\frac{u'}{\sqrt{u^2-1}}$ $(u>1)$
$\tanh^{-1} u$	$\frac{u'}{1-u^2}$ $(\vert u \vert < 1)$
$\coth^{-1} u$	$\frac{u'}{1-u^2}$ $(\vert u \vert > 1)$
$\text{sech}^{-1} u$	$-\frac{u'}{u\sqrt{1-u^2}}$ $(0<u<1)$
$\text{csch}^{-1} u$	$-\frac{u'}{\vert u \vert \sqrt{1+u^2}}$ $(u \neq 0)$

Integration Patterns

Integral	Result
$\int \frac{dx}{\sqrt{a^2+x^2}}$	$\sinh^{-1}\frac{x}{a} + C = \ln(x+\sqrt{x^2+a^2}) + C$
$\int \frac{dx}{\sqrt{x^2-a^2}}$	$\cosh^{-1}\frac{x}{a} + C = \ln(x+\sqrt{x^2-a^2}) + C$
$\int \frac{dx}{a^2-x^2}$	$\frac{1}{a}\tanh^{-1}\frac{x}{a} + C$ or $\frac{1}{2a}\ln\left\vert \frac{a+x}{a-x} \right\vert + C$

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Part A: Derivatives of Inverse Trig Functions

Target: 60–90 seconds each. Just apply the formula + chain rule.

Set A1 — Basic Inverse Trig Derivatives (10 problems)

Find $\frac{dy}{dx}$.

1. $y = \sin^{-1}(4x)$
2. $y = \tan^{-1}(x^3)$
3. $y = \cos^{-1}(\sqrt{x})$
4. $y = \sec^{-1}(5x)$
5. $y = \csc^{-1}(e^x)$
6. $y = \cot^{-1}(2x+1)$
7. $y = \sin^{-1}\left(\frac{x}{3}\right)$
8. $y = \tan^{-1}(\ln x)$
9. $y = \cos^{-1}(e^{-x})$
10. $y = \sec^{-1}(x^2+1)$

Score: ___/10

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Set A2 — Chain Rule Heavy (10 problems)

Find $\frac{dy}{dx}$.

11. $y = \sin^{-1}(\cos x)$
12. $y = \tan^{-1}(\sqrt{x^2+1})$
13. $y = \cos^{-1}\left(\frac{1-x}{1+x}\right)$
14. $y = \sin^{-1}(e^{2x})$
15. $y = \sec^{-1}(\tan x)$
16. $y = \cot^{-1}(\sinh x)$
17. $y = \tan^{-1}\left(\frac{1+x}{1-x}\right)$
18. $y = \cos^{-1}(\sin^{-1} x)$ (just set up derivative)
19. $y = x\sin^{-1}(2x)$
20. $y = e^{x}\tan^{-1}(3x)$

Score: ___/10

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Set A3 — Product, Quotient & Implicit (8 problems)

Find $\frac{dy}{dx}$ or $\frac{dy}{dx}$ in terms of $x$ and $y$.

21. $y = \frac{\sin^{-1} x}{\sqrt{1-x^2}}$
22. $y = \ln(x)\cos^{-1}(2x)$
23. $y = \frac{\tan^{-1} x}{x^2+1}$
24. $y = \sin^{-1} x \cdot \tan^{-1} x$
25. $x\sin^{-1} y + y\sin^{-1} x = \frac{\pi}{2}$
26. $y = \frac{e^{2x}}{\cos^{-1}(x^2)}$
27. $y = \sqrt{x}\sec^{-1}(\sqrt{x})$
28. $y = \left(\sin^{-1} x\right)^2$

Score: ___/8

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Part B: Hyperbolic Functions

Target: 60 seconds each for evaluations; 90 seconds each for derivatives/integrals.

Set B1 — Evaluate & Use Identities (8 problems)

29. If $\sinh x = \frac{3}{4}$, find $\cosh x$ and $\tanh x$.

30. If $\cosh x = \frac{5}{3}$ and $x > 0$, find $\sinh x$ and $\text{sech } x$.

31. Simplify: $\cosh(\ln 2)$.

32. Simplify: $\tanh(\ln 3)$.

33. Show that $\sinh(2x) = 2\sinh x \cosh x$ using exponential definitions.

34. Solve for $x$: $2\cosh^2 x - 5\sinh x = 1$.

35. If $\sinh x = t$, express $\cosh(2x)$ in terms of $t$.

36. Prove: $\cosh x + \sinh x = e^x$.

Score: ___/8

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Set B2 — Hyperbolic Derivatives (10 problems)

Find $\frac{dy}{dx}$.

37. $y = \sinh(3x+2)$
38. $y = \cosh(x^2)$
39. $y = \tanh(\sqrt{x})$
40. $y = \text{sech}(e^x)$
41. $y = \ln(\cosh x)$
42. $y = \sinh^3(2x)$
43. $y = x\cosh x - \sinh x$
44. $y = \frac{\sinh x}{1+\cosh x}$
45. $y = \coth(\ln x)$
46. $y = \sqrt{\tanh(x^2)}$

Score: ___/10

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Set B3 — Hyperbolic Integrals (8 problems)

Evaluate.

47. $\int \sinh(4x),dx$
48. $\int \cosh(2x-1),dx$
49. $\int \tanh x,dx$ (Hint: write as $\frac{\sinh x}{\cosh x}$)
50. $\int \sinh^2 x \cosh x,dx$
51. $\int \frac{\text{sech}^2 x}{1+\tanh x},dx$
52. $\int x\sinh(x^2),dx$
53. $\int e^x \sinh x,dx$ (Hint: use exponential definition)
54. $\int_0^{\ln 2} \text{sech } x,dx$ (Hint: multiply by $\frac{\cosh x}{\cosh x}$)

Score: ___/8

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Part C: Inverse Hyperbolic Functions

Target: 90 seconds each. Formula + chain rule, or standard integral pattern.

Set C1 — Derivatives of Inverse Hyperbolic (10 problems)

Find $\frac{dy}{dx}$.

55. $y = \sinh^{-1}(3x)$
56. $y = \cosh^{-1}(x^2)$
57. $y = \tanh^{-1}(\sin x)$
58. $y = \text{sech}^{-1}(x^3)$
59. $y = \coth^{-1}(e^x)$
60. $y = x\sinh^{-1}(x)$
61. $y = \ln(\cosh^{-1} x)$
62. $y = \frac{\tanh^{-1} x}{x}$
63. $y = \sinh^{-1}(\cos x)$
64. $y = \cosh^{-1}\left(\frac{x}{a}\right)$ (general formula)

Score: ___/10

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Set C2 — Integrals → Inverse Hyperbolic / Log Forms (10 problems)

Evaluate.

65. $\int \frac{dx}{\sqrt{4+x^2}}$
66. $\int \frac{dx}{\sqrt{x^2-9}}$
67. $\int \frac{dx}{25-x^2}$
68. $\int \frac{dx}{\sqrt{1+4x^2}}$
69. $\int \frac{dx}{x\sqrt{1-x^2}}$ (Hint: substitute $u = \frac{1}{x}$ or recognize pattern)
70. $\int \frac{dx}{\sqrt{x^2+6x+13}}$ (Complete the square first)
71. $\int \frac{dx}{\sqrt{9x^2-16}}$
72. $\int \frac{dx}{3-5x^2}$
73. $\int \frac{e^x}{\sqrt{e^{2x}+1}},dx$ (Hint: $u = e^x$)
74. $\int_0^{1/2} \frac{dx}{1-4x^2}$

Score: ___/10

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Set C3 — Rapid Mixed Fire (8 problems)

Find $\frac{dy}{dx}$ or evaluate the integral.

75. $y = \sin^{-1}(\tanh x)$
76. $\int \sinh x \cos^{-1}(\cosh x),dx$ (Hint: substitution)
77. $y = \cosh^{-1}(\sec x)$
78. $\int \frac{dx}{\sqrt{(x+1)^2+4}}$
79. $y = \tanh^{-1}\left(\frac{2x}{1+x^2}\right)$ (simplify first)
80. $\int \frac{dx}{\sqrt{4x^2+12x+5}}$ (factor under root)
81. $y = \sinh^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$
82. $\int_0^{\ln 3} \frac{dx}{\sqrt{1+e^{2x}}}$ (Hint: $u = e^{-x}$)

Score: ___/8

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Final Scorecard

Part	Sets	Problems	Raw Score
A — Inverse Trig Derivatives	A1, A2, A3	28	___/28
B — Hyperbolic Functions	B1, B2, B3	26	___/26
C — Inverse Hyperbolic	C1, C2, C3	28	___/28
TOTAL		82	___/82

Proficiency Benchmarks

• 60/82 (73%) — Proficient. You can handle standard exam problems.
• 70/82 (85%) — Solid. Fast and accurate.
• 76/82 (93%) — Exam-ready. Any mistake is a careless slip.

Speed Benchmarks

• <2 hours: Excellent mechanical fluency.
• 2–2.5 hours: Good. Review missed patterns.
• >3 hours: Drill the specific sets you scored lowest on again tomorrow.

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Error Log Template

After grading, list every wrong problem number with a one-word reason:

Problem	Reason
e.g. 14	sign error
e.g. 47	forgot chain rule factor

Re-solve all wrong problems immediately with notes, then again in 24 hours without notes.

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Related Resources

• FAC1004 Tutorial 7 — Derivatives of Inverse Trig & Hyperbolic Functions
• FAC1004 Tutorial 8 — Hyperbolic Functions
• FAC1004 Tutorial 9 — Inverse Hyperbolic Functions
• FAC1004 - Advanced Mathematics II (Computing)
• Inverse Trigonometric Functions — concept page
• Hyperbolic Functions — concept page

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