DerivativeCalculus.com

Differentiation Rules Cheat Sheet

Your comprehensive guide to all essential derivative rules and formulas. Perfect for quick reference and exam preparation.

🔢 Basic Differentiation Rules
⚡ Power Rule
The most fundamental rule for differentiating polynomial terms.
d/dx [xⁿ] = n·xⁿ⁻¹
Example:
d/dx [x⁵] = 5x⁴
🔢 Constant Rule
The derivative of any constant is zero.
d/dx [c] = 0
Example:
d/dx [7] = 0
✖️ Constant Multiple Rule
Constants can be factored out of derivatives.
d/dx [c·f(x)] = c·f'(x)
Example:
d/dx [5x³] = 5·3x² = 15x²
➕ Sum Rule
The derivative of a sum is the sum of derivatives.
d/dx [f(x) + g(x)] = f'(x) + g'(x)
Example:
d/dx [x² + 3x] = 2x + 3
➖ Difference Rule
The derivative of a difference is the difference of derivatives.
d/dx [f(x) - g(x)] = f'(x) - g'(x)
Example:
d/dx [x³ - 2x] = 3x² - 2
🚀 Advanced Differentiation Rules
✖️ Product Rule
For differentiating products of two functions.
d/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)
Example:
d/dx [x²·sin(x)] = 2x·sin(x) + x²·cos(x)
➗ Quotient Rule
For differentiating quotients of two functions.
d/dx [f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]²
Example:
d/dx [sin(x)/x] = [x·cos(x) - sin(x)] / x²
🔗 Chain Rule
For differentiating composite functions.
d/dx [f(g(x))] = f'(g(x))·g'(x)
Example:
d/dx [sin(x²)] = cos(x²)·2x
⚡🔗 Generalized Power Rule
Power rule combined with chain rule.
d/dx [uⁿ] = n·uⁿ⁻¹·u'
Example:
d/dx [(x² + 1)³] = 3(x² + 1)²·2x
🌟 Transcendental Functions
📈 Exponential Functions
Derivatives of exponential functions with base e.
d/dx [eˣ] = eˣ
d/dx [eᵘ] = eᵘ·u'
Example:
d/dx [e²ˣ] = 2e²ˣ
📊 Logarithmic Functions
Derivatives of natural logarithmic functions.
d/dx [ln(x)] = 1/x
d/dx [ln(u)] = u'/u
Example:
d/dx [ln(x²)] = 2x/x² = 2/x
📐 Trigonometric Functions
Basic trigonometric function derivatives.
d/dx [sin(x)] = cos(x)
d/dx [cos(x)] = -sin(x)
d/dx [tan(x)] = sec²(x)
Example:
d/dx [sin(2x)] = 2cos(2x)
🔄 Inverse Trigonometric
Derivatives of inverse trigonometric functions.
d/dx [arcsin(x)] = 1/√(1-x²)
d/dx [arctan(x)] = 1/(1+x²)
d/dx [arcsec(x)] = 1/(|x|√(x²-1))
Example:
d/dx [arctan(3x)] = 3/(1+9x²)
⚠️ Important: Remember the Domain Restrictions

• ln(x) is only defined for x > 0

• √x is only defined for x ≥ 0

• 1/x is undefined at x = 0

• Trigonometric inverses have restricted domains

Always check the domain of your original function before differentiating!

📋 Quick Reference Table
xⁿ
n·xⁿ⁻¹
eˣ
eˣ
aˣ
aˣ·ln(a)
ln(x)
1/x
sin(x)
cos(x)
cos(x)
-sin(x)
tan(x)
sec²(x)
cot(x)
-csc²(x)
sec(x)
sec(x)tan(x)
csc(x)
-csc(x)cot(x)
arcsin(x)
1/√(1-x²)
arctan(x)
1/(1+x²)
🧠 Memory Tips & Mnemonics
🎵 Trigonometric Derivatives Song

"Sine goes to cosine, cosine goes to minus sine,
Add one to the power, reduce the power by one!"

🔤 Product Rule Acronym

D(First) × S(econd) + F(irst) × D(Second)
"Derivative of First times Second, plus First times Derivative of Second"

👇 Quotient Rule Rhyme

"Low D-High minus High D-Low,
Square the bottom and away you go!"

⛓️ Chain Rule Visualization

Think of it as "outside derivative, keep inside the same,
multiply by inside derivative" - like peeling an onion!

⚡ Exponential Memory Hook

eˣ is the only function that equals its own derivative -
"eˣ is special, it's its own rate of change!"

🎯 Practice Makes Perfect

• Practice 10-15 derivative problems daily for two weeks

• Start with basic rules, then combine rules

• Use our calculators to check your work

• Create flashcards for formulas you find difficult

• Time yourself to build speed for exams