[Leejohnston]

Derivative & Integral Formula Sheet — GCSE / A-Level Friendly

A clear, simple reference sheet covering the essential differentiation and integration formulas. 
Perfect for calculus beginners, A-Level students, physics learners, and anyone doing mathematical modelling.

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1. Basic Differentiation Rules

Rule 1 — Power Rule 
If y = x[sup]n[/sup], then:

dy/dx = n·x[sup]n−1[/sup]

Examples: 
• d/dx (x[sup]3[/sup]) = 3x[sup]2[/sup] 
• d/dx (x[sup]7[/sup]) = 7x[sup]6[/sup] 
• d/dx (x[sup]1[/sup]) = 1 

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Rule 2 — Constant Rule 
d/dx (k) = 0 
(k is any constant)

Examples: 
• d/dx (5) = 0 
• d/dx (100) = 0 

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Rule 3 — Constant Multiplier 
d/dx (k·f(x)) = k·f’(x)

Example: 
d/dx (4x[sup]3[/sup]) = 4 · 3x[sup]2[/sup] = 12x[sup]2[/sup]

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Rule 4 — Sum & Difference Rule 
Differentiate each term separately.

Examples: 
• d/dx (3x[sup]2[/sup] + 4x + 7) = 6x + 4 
• d/dx (5x[sup]3[/sup] − 2x) = 15x[sup]2[/sup] − 2 

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2. Basic Integration Rules

Integration is the opposite of differentiation.

Rule 1 — Reverse Power Rule 
∫ x[sup]n[/sup] dx = x[sup]n+1[/sup] / (n+1) + C 
(as long as n ≠ −1)

Examples: 
• ∫ x[sup]2[/sup] dx = x[sup]3[/sup]/3 + C 
• ∫ x[sup]5[/sup] dx = x[sup]6[/sup]/6 + C 
• ∫ x dx = x[sup]2[/sup]/2 + C 

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Rule 2 — Constant Rule 
∫ k dx = kx + C

Examples: 
• ∫ 5 dx = 5x + C 
• ∫ 12 dx = 12x + C 

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Rule 3 — Sum & Difference Rule 
Integrate each term separately.

Example: 
∫ (3x[sup]2[/sup] − 4x + 6) dx 
= x[sup]3[/sup] − 2x[sup]2[/sup] + 6x + C 

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3. Special Cases

Derivative of e[sup]x[/sup]: 
d/dx (e[sup]x[/sup]) = e[sup]x[/sup]

Integral of e[sup]x[/sup]: 
∫ e[sup]x[/sup] dx = e[sup]x[/sup] + C

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Derivative of ln(x): 
d/dx (ln x) = 1/x

Integral of 1/x: 
∫ (1/x) dx = ln|x| + C 

(This is the exception to the power rule.)

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4. Basic Trig Differentiation (A-Level)

d/dx (sin x) = cos x 
d/dx (cos x) = −sin x 
d/dx (tan x) = sec[sup]2[/sup] x

Integrals:

∫ sin x dx = −cos x + C 
∫ cos x dx = sin x + C 
∫ sec[sup]2[/sup] x dx = tan x + C

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5. Applied to Physics

Velocity = derivative of displacement 
Acceleration = derivative of velocity 

[x → dx/dt → d²x/dt²]

Integrating reverses the process.

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6. Common Mistakes

❌ Forgetting +C in integrals 
✔ Always include the constant

❌ Using power rule on 1/x 
✔ Use ln|x| 

❌ Dropping negative signs in trig 
✔ Remember: cos → −sin 

❌ Mixing up differentiation & integration 
✔ Check the direction carefully 

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Summary

Key rules to memorise:

Differentiation: 
• x[sup]n[/sup] → n·x[sup]n−1[/sup] 
• constants → 0 
• term-by-term 

Integration: 
• x[sup]n[/sup] → x[sup]n+1[/sup]/(n+1) 
• constants → kx 
• term-by-term 
• +C always included 

Master these and you’ll be comfortable with all core calculus work.