Differentiation And Integration (Copy)
DIFFERENTIATION AND INTEGRATION — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Differentiation | dy/dx | Treating as fraction always |
| Power Rule | d/dx(xⁿ) = nxⁿ⁻¹ | Forgetting to reduce power |
| Constant Rule | d/dx(c) = 0 | Writing constant again |
| Constant Multiple Rule | d/dx(cf(x)) = c(d/dx f(x)) | Ignoring constant |
| Sum Rule | d/dx(u + v) = du/dx + dv/dx | Combining terms incorrectly |
| Difference Rule | d/dx(u − v) = du/dx − dv/dx | Sign mistakes |
| Product Rule | d/dx(uv) = u(dv/dx) + v(du/dx) | Missing one term |
| Quotient Rule | d/dx(u/v) = (v du/dx − u dv/dx)/v² | Wrong order in numerator |
| Chain Rule | d/dx[f(g(x))] = f'(g(x))g'(x) | Forgetting inner derivative |
| Exponential Derivative | d/dx(eˣ) = eˣ | Multiplying by x |
| Natural Log Derivative | d/dx(lnx) = 1/x | Forgetting denominator |
| Trig Derivative | d/dx(sinx) = cosx | Wrong trig function |
| Trig Derivative | d/dx(cosx) = −sinx | Forgetting negative sign |
| Trig Derivative | d/dx(tanx) = sec²x | Writing secx |
| Stationary Point | dy/dx = 0 | Forgetting to solve |
| Increasing Function | dy/dx > 0 | Mixing signs |
| Decreasing Function | dy/dx < 0 | Mixing signs |
| Maximum Point | Gradient changes + to − | Wrong sign order |
| Minimum Point | Gradient changes − to + | Wrong sign order |
| Tangent Gradient | Gradient at a point | Using average gradient |
| Normal Gradient | m₁m₂ = −1 | Forgetting negative reciprocal |
| Tangent Equation | y − y₁ = m(x − x₁) | Wrong point substitution |
| Approximation Formula | y ≈ f(a) + f'(a)(x − a) | Wrong substitution |
| Rate of Change | dy/dx | Confusing variables |
| Second Derivative | d²y/dx² | Treating same as first derivative |
| Concave Up | d²y/dx² > 0 | Mixing inequality |
| Concave Down | d²y/dx² < 0 | Mixing inequality |
| Point of Inflection | d²y/dx² changes sign | Using dy/dx only |
| Integration | ∫f(x)dx | Forgetting constant |
| Constant of Integration | + C | Omitting completely |
| Reverse Power Rule | ∫xⁿdx = xⁿ⁺¹/(n + 1) + C | Forgetting +1 |
| Special Integration Rule | ∫1/x dx = lnx + C | Using power rule |
| Exponential Integration | ∫eˣdx = eˣ + C | Adding denominator |
| Trig Integration | ∫cosx dx = sinx + C | Wrong sign |
| Trig Integration | ∫sinx dx = −cosx + C | Forgetting negative |
| Definite Integral | ∫ₐᵇf(x)dx = F(b) − F(a) | Wrong substitution order |
| Area Under Curve | Use definite integral | Using coordinates only |
| Area Below x-axis | Take absolute value | Leaving negative area |
| Velocity from Acceleration | Integrate acceleration | Differentiating instead |
| Displacement from Velocity | Integrate velocity | Differentiating instead |
| Integration by Substitution | Substitute carefully | Forgetting dx conversion |
| Differential Equation | Solve by integrating | Leaving derivative form |
| Initial Conditions | Substitute to find C | Leaving unknown constant |
| Gradient Function | First derivative | Using original function |
| Area Between Curves | Upper − lower | Reversing subtraction |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK DIFFERENTIATION RULES TABLE
| Function | Derivative |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| c | 0 |
| eˣ | eˣ |
| lnx | 1/x |
| sinx | cosx |
| cosx | −sinx |
| tanx | sec²x |
QUICK INTEGRATION RULES TABLE
| Function | Integral |
|---|---|
| xⁿ | xⁿ⁺¹/(n + 1) + C |
| 1/x | lnx + C |
| eˣ | eˣ + C |
| cosx | sinx + C |
| sinx | −cosx + C |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK STATIONARY POINT RULES
| Situation | Condition |
|---|---|
| Stationary Point | dy/dx = 0 |
| Maximum Point | Gradient changes + to − |
| Minimum Point | Gradient changes − to + |
| Point of Inflection | d²y/dx² changes sign |
QUICK TANGENT & NORMAL RULES
| Topic | Formula |
|---|---|
| Tangent Equation | y − y₁ = m(x − x₁) |
| Normal Gradient | −1/m |
| Perpendicular Condition | m₁m₂ = −1 |
QUICK AREA RULES
| Situation | Formula |
|---|---|
| Area Under Curve | ∫ₐᵇf(x)dx |
| Area Between Curves | ∫(top − bottom)dx |
| Definite Integral | F(b) − F(a) |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Differentiate powers | Bring power forward |
| Integrate powers | Add 1 to power first |
| Stationary point | Set dy/dx = 0 |
| Tangent question | Differentiate first |
| Normal question | Negative reciprocal |
| Definite integral | Upper limit minus lower |
| Area below axis | Use positive area |
| Chain rule | Differentiate outside then inside |
| Product rule | First-second plus second-first |
| Quotient rule | Bottom-top minus top-bottom |
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Forgetting + C | Integration incomplete |
| Wrong power after integration | Entire answer incorrect |
| Missing chain rule | Wrong derivative |
| Product rule missing term | Method lost |
| Quotient rule order error | Sign mistakes |
| Wrong trig derivative | Formula marks lost |
| Forgetting limits substitution | Incorrect area |
| Negative area left unchanged | Wrong final answer |
| Arithmetic sign mistakes | Accuracy lost |
| Not simplifying final answer | Method marks lost |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.