Comprehensive Calculus Reference: Derivatives and Integrals
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Table of Derivatives
| Role | Derived Function | Role | Derived Function |
|---|---|---|---|
| Y = k | Y' = 0 | Y = x | Y' = 1 |
| Y = u + v + w | Y' = u' + v' + w' | Y = u · v | Y' = u · v' + u' · v |
| Y = u/v | Y' = (v · u' - v' · u) / v² | Y = log_b u | Y' = u' / (u · ln b) |
| Y = uⁿ | Y' = n · uⁿ⁻¹ · u' | Y = ln u | Y' = u' / u |
| Y = kᵘ | Y' = u' · kᵘ · ln k | Y = eᵘ | Y' = u' · eᵘ |
| Y = sin u | Y' = u' · cos u | Y = csc u | Y' = -u' · cot u · csc u |
| Y = cos u | Y' = -u' · sin u | Y = sec u | Y' = u' · tan u · sec u |
| Y = tan u | Y' = u' · sec² u | Y = cot u | Y' = -u' · csc² u |
Table of Integrals
| Role | Integral Result | Role | Integral Result |
|---|---|---|---|
| ∫ k du | k · u + C | ∫ uⁿ du | uⁿ⁺¹ / (n + 1) + C |
| ∫ du / u | ln |u| + C | ∫ eᵘ du | eᵘ + C |
| ∫ sin u du | -cos u + C | ∫ cos u du | sin u + C |
| ∫ tan u du | ln |sec u| + C | ∫ cot u du | ln |sin u| + C |
| ∫ sec² u du | tan u + C | ∫ csc² u du | -cot u + C |
Note: In all formulas, C represents the constant of integration. u, v, and w are functions of x.