How to Memorize AP Calculus Derivatives and Integrals Quickly | RevisionDojo

Derivatives and integrals are the lifeblood of AP Calculus AB & BC. If you can recall them instantly, you’ll save precious time on multiple-choice and free-response questions — and reduce careless mistakes.

In this RevisionDojo guide, you’ll learn:

• The must-know derivative and integral formulas for the AP exam
• Fast memorization techniques
• Mnemonics for tricky functions
• How to review so you never forget them under test pressure

📚 Must-Know Derivative Formulas

Power Rule:

ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1}

Exponential & Logarithmic:

ddxex=ex\frac{d}{dx}e^x = e^xddxln⁡∣x∣=1x\frac{d}{dx}\ln|x| = \frac{1}{x}

Trigonometric:

ddxsin⁡x=cos⁡x\frac{d}{dx}\sin x = \cos xddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x = -\sin xddxtan⁡x=sec⁡2x\frac{d}{dx}\tan x = \sec^2 x

Inverse Trig:

ddxsin⁡−1x=11−x2\frac{d}{dx}\sin^{-1}x = \frac{1}{\sqrt{1 - x^2}}ddxtan⁡−1x=11+x2\frac{d}{dx}\tan^{-1}x = \frac{1}{1 + x^2}

📚 Must-Know Integral Formulas

Power Rule for Integrals:

∫xndx=xn+1n+1+C(n≠−1)\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

Exponential & Logarithmic:

∫exdx=ex+C\int e^x dx = e^x + C∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln|x| + C

Trigonometric:

∫sin⁡xdx=−cos⁡x+C\int \sin x dx = -\cos x + C∫cos⁡xdx=sin⁡x+C\int \cos x dx = \sin x + C∫sec⁡2xdx=tan⁡x+C\int \sec^2 x dx = \tan x + C

Inverse Trig:

∫11−x2dx=sin⁡−1x+C\int \frac{1}{\sqrt{1 - x^2}} dx = \sin^{-1}x + C∫11+x2dx=tan⁡−1x+C\int \frac{1}{1 + x^2} dx = \tan^{-1}x + C

⚡ Quick Memorization Techniques

1. Chunking

Group formulas into categories: power rules, exponentials, trig, inverse trig. Memorize one category at a time.

2. Spaced Repetition

Review daily for the first week, then every 2–3 days. Tools like Anki or Quizlet help automate this.

3. Write to Remember

Physically write each formula five times — research shows handwriting strengthens recall.

4. Recite Out Loud

Say formulas aloud while practicing problems. Hearing and speaking boosts memory.

🎯 Mnemonics for Tricky Formulas

• Derivative of sin = cos → Think “sin grows into cos.”
• Derivative of cos = -sin → “Cos falls into sin — negative because it’s falling.”
• Derivative of tan = sec² → Imagine tan needs “security” (sec²) to stay upright.

📝 Practice Integration With Differentiation

If you forget an integral, recall the derivative instead. Example: If ddxsin⁡x=cos⁡x\frac{d}{dx}\sin x = \cos x, then ∫cos⁡xdx=sin⁡x+C\int \cos x dx = \sin x + C.

🔄 Daily Drill Routine (RevisionDojo Method)

Morning: Write all derivative formulas from memory.
Afternoon: Write all integral formulas from memory.
Evening: Do 5 practice questions where you must choose the correct derivative or integral without a calculator.

⚠️ Common Mistakes to Avoid

• Forgetting the + C in indefinite integrals
• Mixing up sec⁡2x\sec^2 x (derivative of tan) with −csc⁡2x-\csc^2 x (derivative of cot)
• Confusing sin⁡−1x\sin^{-1}x (inverse sine) with 1sin⁡x\frac{1}{\sin x} (cosecant)

🧭 Final Advice from RevisionDojo

The AP Calculus exam rewards speed + accuracy. The faster you can recall derivatives and integrals, the more time you have for complex multi-step problems.
Memorize them in categories, use spaced repetition, and practice retrieval under timed conditions — and you’ll be exam-ready in no time.