Calculus is the largest Pure Maths topic — from basic differentiation/integration through advanced techniques (chain, product, quotient, by parts, substitution, partial fractions) to differential equations and numerical methods.
Basic: d/dx(xⁿ) = nxⁿ⁻¹. d/dx(sinx) = cosx, d/dx(cosx) = −sinx, d/dx(tanx) = sec²x. d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x. Chain rule: dy/dx = dy/du × du/dx. Product rule: d/dx(uv) = u(dv/dx) + v(du/dx). Quotient rule: d/dx(u/v) = (v(du/dx) − u(dv/dx))/v². Implicit: differentiate every term, dy/dx appears when differentiating y terms (chain rule). Parametric: dy/dx = (dy/dt)/(dx/dt). Applications: tangents, normals, stationary points, rates of change, connected rates.
Reverse of differentiation: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠−1). ∫1/x dx = ln|x| + C. ∫eˣ dx = eˣ + C. ∫sinx dx = −cosx + C. Integration by substitution: choose u, replace dx. Integration by parts: ∫u dv = uv − ∫v du (LIATE rule). Partial fractions for rational integrals. Trapezium rule: ∫ ≈ (h/2)(y₀ + 2y₁ + 2y₂ + … + yₙ). Differential equations: separate variables, integrate both sides. Use initial conditions to find C. Area under curve: ∫ₐᵇ y dx. Volume of revolution: V = π∫ₐᵇ y² dx.
Decision tree: (1) Standard integral (xⁿ, sinx, eˣ, 1/x)? → Integrate directly. (2) Is it a product of two different types (e.g., xeˣ, x sinx)? → Integration by parts (LIATE: Logarithmic, Inverse trig, Algebraic, Trig, Exponential — put earlier type as u). (3) Is there a composite function (e.g., sin(3x+1), e²ˣ)? → Substitution (or reverse chain rule for simple cases). (4) Is it a rational function P(x)/Q(x)? → If degree of P ≥ degree of Q, do polynomial division first. Then use partial fractions to split the remainder. (5) Trigonometric identities: use sin²x = (1-cos2x)/2 etc. to simplify before integrating. Practice is key — recognising the type becomes instinctive with experience.
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