1. Fundamental Differentiation Operational Rules

Power Rule: d/dx(xⁿ) = n·xⁿ⁻¹. Constant Multiple Rule: d/dx[c·f(x)] = c·f'(x). Sum/Difference Rule: d/dx[f(x) ± g(x)] = f'(x) ± g'(x). Product Rule: d/dx[u·v] = u·(dv/dx) + v·(du/dx). Quotient Rule: d/dx[u/v] = [v·(du/dx) - u·(dv/dx)] / v². Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x).

2. Standard Trigonometric & Transcendental Derivatives

Trigonometric: d/dx(sin x) = cos x; d/dx(cos x) = -sin x; d/dx(tan x) = sec² x; d/dx(cot x) = -csc² x; d/dx(sec x) = sec x tan x; d/dx(csc x) = -csc x cot x (notice that all co-functions carry negative derivatives). Exponential & Logarithmic: d/dx(eˣ) = eˣ; d/dx(aˣ) = aˣ ln a; d/dx(ln |x|) = 1/x.

3. Standard Indefinite Integrals Reference Table

Basic integrals (add constant +C): ∫ xⁿ dx = (xⁿ⁺¹)/(n + 1) [for n ≠ -1]; ∫ (1/x) dx = ln |x|; ∫ eˣ dx = eˣ; ∫ aˣ dx = aˣ / ln a. Trigonometric integrals: ∫ sin x dx = -cos x; ∫ cos x dx = sin x; ∫ sec² x dx = tan x; ∫ csc² x dx = -cot x; ∫ sec x tan x dx = sec x; ∫ (1 / √(1 - x²)) dx = sin⁻¹ x; ∫ (1 / (1 + x²)) dx = tan⁻¹ x.

4. Integration by Parts & The ILATE Priority Rule

Integration by parts formula: ∫ u dv = u·v - ∫ v du. To choose the first function u, use the ILATE priority order: I (Inverse trigonometric), L (Logarithmic), A (Algebraic), T (Trigonometric), E (Exponential). The function appearing earlier in ILATE is designated as u, making the remaining term dv.

5. Fundamental Limit Approximations

Key standard limits as x → 0: lim (sin x / x) = 1 (where x is in radians); lim (tan x / x) = 1; lim (1 - cos x)/x² = 1/2; lim (eˣ - 1)/x = 1; lim [ln(1 + x)]/x = 1. For indeterminate forms (0/0 or ∞/∞), L'Hôpital's Rule allows differentiating numerator and denominator separately: lim [f(x)/g(x)] = lim [f'(x)/g'(x)].