1. Differentiation
1.1. Dy/Dx
1.2. Chain Rule d/dx[f(g(x))]=f'(g(x))g'(x)
1.3. Quotient Rule d/dx[f(x)/g(x)]= g(x)f'(x)-f(x)g'9(x)/[g(x)]^2
1.4. Constant Rule d/dx[c]=0
2. Modulus Inequalities
2.1. properties
2.1.1. |a| ≤ b => -b ≤ a ≤ b
2.1.2. |a| ≥ b => a ≤ -b OR a ≥ b
2.1.3. |a| ≥ |b| => a² ≥ b²
2.1.4. |a| ≤ |b| => a² ≤ b²
3. Partial Fractions
3.1. type1
3.1.1. rational fraction: (px + q)/(ax + b)
3.1.2. partial fraction: A/(ax + b)
3.2. type 2
3.2.1. rational fraction: (px + q)/(ax + b)n
3.2.2. partial fraction: A1/(ax + b) + .......... An/(ax + b)n
3.3. type 3
3.3.1. rational fraction: (px2 + qx + r)/(ax2 + bx + c)
3.3.2. partial fraction: (Ax + B)/(ax2 + bx + c)
3.4. type 4
3.4.1. rational fraction: (px2 + qx + r)/(ax2 + bx + c)n
3.4.2. partial fraction: (A1x + B1)/(ax2 + bx + c) + ...(Anx + Bn)/(ax2 + bx + c)n
3.5. uses
3.5.1. useful for finding binomial approximations
3.5.2. useful for integrating a rational function
4. Polynomials
4.1. synthetic division
4.2. long division
5. Trigonometry
5.1. The Cosecant, Secant and Cotangent Ratios
5.1.1. cosec θ =1/sinθ
5.1.2. sec θ = 1/cosθ
5.1.3. cot θ = cos θ/sinθ or 1/tan θ
5.2. Compound Angle Formulae
5.2.1. sin(A+B) = sin A cos B + cos A sin B
5.2.2. sin(A-B) = sin A cos B - cos A sin B
5.2.3. cos(A+B) = cos A cos B + sin A sin B
5.2.4. cos(A-B) = cos A cos B - sin A sin B
5.2.5. tan(A+B) = tan A + tan B/1- tan A tan B
5.2.6. tan(A-B) = tan A - tan B/1+ tan A tan B
5.3. Double Angle Formulae
5.3.1. sin 2A = 2 sin A cos A
5.3.2. cos 2 A = cos^2 A - sin^2 A = 2cos^2 A -1 = 1 - 2sin^2 A
5.3.3. tan^2 A = 2 tan A/ 1- tan^2 A
5.4. Further Trigonometric Identities
5.4.1. 1+ tan^2x = sec^2x
5.4.2. 1 + cot^2x = cosec^2x
5.5. Expressing asinθ + bcosθ in the form R sin(θ+α) or R cos(θ+α)
5.5.1. R = √a^2 + b^2, tan α = b/α
5.5.2. max. value of asinθ + bcosθ is √a^2 + b^2 when sin(θ+α) = 1
5.5.3. min. value of asinθ + bcosθ is -√a^2 + b^2 when sin(θ+α) = -1
6. Logarithm and Exponential Functions
6.1. a^b = y can be annotated as loga y = b
6.2. loge x = ln x log10 x = lg x
6.3. Laws of Logarithm
6.3.1. loga x + loga y = loga (xy)
6.3.2. loga x – loga y = loga (x/y)
6.3.3. loga x^n = n loga x
6.4. Natural logarithm (ln x)
6.4.1. derivative of ln x is 1/x
6.4.2. integral of 1/x is ln x + c