Derivative Rules Cheat Sheet
Your complete quick reference guide to all derivative rules, formulas, and techniques. Perfect for studying, exams, and homework. Print-friendly and organized by category.
📐 Basic Derivative Rules
Constant Rule
The derivative of any constant is zero.
d/dx[7] = 0
d/dx[-15] = 0
d/dx[π] = 0
Power Rule
Multiply by the exponent and reduce the exponent by 1.
d/dx[x³] = 3x²
d/dx[x⁵] = 5x⁴
d/dx[x] = 1
d/dx[√x] = d/dx[x^(1/2)] = (1/2)x^(-1/2) = 1/(2√x)
Constant Multiple Rule
Constants can be pulled out of the derivative.
d/dx[5x³] = 5·3x² = 15x²
d/dx[-2x⁴] = -2·4x³ = -8x³
d/dx[πx²] = π·2x = 2πx
Sum/Difference Rule
Take the derivative of each term separately.
d/dx[x³ + x²] = 3x² + 2x
d/dx[5x⁴ - 3x²] = 20x³ - 6x
d/dx[x⁵ + 2x³ - 7x] = 5x⁴ + 6x² - 7
🔗 Advanced Rules
Product Rule
Mnemonic: "First times derivative of second, plus second times derivative of first"
d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x)
Quotient Rule
Mnemonic: "Low d-high minus high d-low, over the square of what's below"
d/dx[x²/sin(x)] = [2x·sin(x) - x²·cos(x)] / sin²(x)
Chain Rule
Mnemonic: "Derivative of outside times derivative of inside"
d/dx[(x² + 1)⁵] = 5(x² + 1)⁴·(2x)
d/dx[sin(x²)] = cos(x²)·(2x)
d/dx[e^(3x)] = e^(3x)·(3)
Trigonometric Functions
| Function | Derivative | Example |
|---|---|---|
| sin(x) | cos(x) | d/dx[sin(3x)] = 3cos(3x) |
| cos(x) | -sin(x) | d/dx[cos(2x)] = -2sin(2x) |
| tan(x) | sec²(x) | d/dx[tan(x)] = sec²(x) |
| cot(x) | -csc²(x) | d/dx[cot(x)] = -csc²(x) |
| sec(x) | sec(x)tan(x) | d/dx[sec(x)] = sec(x)tan(x) |
| csc(x) | -csc(x)cot(x) | d/dx[csc(x)] = -csc(x)cot(x) |
Co-functions (cos, cot, csc) have negative derivatives. Regular functions (sin, tan, sec) have positive derivatives (except cos which is co-function).
Inverse Trigonometric Functions
| Function | Derivative |
|---|---|
| arcsin(x) or sin⁻¹(x) | 1/√(1 - x²) |
| arccos(x) or cos⁻¹(x) | -1/√(1 - x²) |
| arctan(x) or tan⁻¹(x) | 1/(1 + x²) |
| arccot(x) or cot⁻¹(x) | -1/(1 + x²) |
| arcsec(x) or sec⁻¹(x) | 1/(|x|√(x² - 1)) |
| arccsc(x) or csc⁻¹(x) | -1/(|x|√(x² - 1)) |
Exponential & Logarithmic Functions
| Function | Derivative | Example |
|---|---|---|
| eˣ | eˣ | d/dx[e^(2x)] = 2e^(2x) |
| aˣ | aˣ·ln(a) | d/dx[2ˣ] = 2ˣ·ln(2) |
| ln(x) | 1/x | d/dx[ln(3x)] = 1/x |
| log_a(x) | 1/(x·ln(a)) | d/dx[log₂(x)] = 1/(x·ln(2)) |
| ln|x| | 1/x | d/dx[ln|x|] = 1/x |
eˣ is the only function that is its own derivative! This makes it incredibly useful in calculus and differential equations.
Hyperbolic Functions
| Function | Derivative |
|---|---|
| sinh(x) | cosh(x) |
| cosh(x) | sinh(x) |
| tanh(x) | sech²(x) |
| coth(x) | -csch²(x) |
| sech(x) | -sech(x)tanh(x) |
| csch(x) | -csch(x)coth(x) |
Hyperbolic derivatives are similar to trig derivatives, but WITHOUT the negative signs for sinh and cosh (unlike sin and cos)!
🎯 Special Differentiation Techniques
Implicit Differentiation
Used when y cannot be easily isolated. Differentiate both sides with respect to x and multiply by dy/dx when differentiating y terms.
x² + y² = 25
2x + 2y(dy/dx) = 0
dy/dx = -x/y
Logarithmic Differentiation
Useful for products, quotients, or powers. Take ln of both sides, then differentiate.
y = xˣ
ln(y) = x·ln(x)
(1/y)(dy/dx) = ln(x) + 1
dy/dx = xˣ(ln(x) + 1)
Parametric Differentiation
For parametric equations x = f(t) and y = g(t).
x = t², y = t³
dx/dt = 2t, dy/dt = 3t²
dy/dx = 3t²/2t = 3t/2
🔢 Higher Order Derivatives
First Derivative: f'(x), dy/dx, y', Df(x)
Second Derivative: f''(x), d²y/dx², y'', D²f(x)
Third Derivative: f'''(x), d³y/dx³, y''', D³f(x)
nth Derivative: f⁽ⁿ⁾(x), dⁿy/dxⁿ, y⁽ⁿ⁾, Dⁿf(x)
Common Higher Order Derivatives
| Function | f'(x) | f''(x) | f'''(x) |
|---|---|---|---|
| xⁿ | nxⁿ⁻¹ | n(n-1)xⁿ⁻² | n(n-1)(n-2)xⁿ⁻³ |
| eˣ | eˣ | eˣ | eˣ |
| sin(x) | cos(x) | -sin(x) | -cos(x) |
| cos(x) | -sin(x) | -cos(x) | sin(x) |
| ln(x) | 1/x | -1/x² | 2/x³ |
🧠 Memory Aids & Mnemonics
⚠️ Common Mistakes to Avoid
Wrong: d/dx[sin(x²)] = cos(x²)
Right: d/dx[sin(x²)] = cos(x²)·(2x)
Wrong: d/dx[x·sin(x)] = 1·cos(x)
Right: d/dx[x·sin(x)] = 1·sin(x) + x·cos(x)
Wrong: d/dx[f/g] = (f'g + fg')/g²
Right: d/dx[f/g] = (f'g - fg')/g² (MINUS in numerator!)
Wrong: d/dx[eˣ] = x·e^(x-1)
Right: d/dx[eˣ] = eˣ (e is NOT a variable!)
Wrong: d/dx[log(x)] = 1/x
Right: d/dx[ln(x)] = 1/x (only for natural log!)
For log₁₀(x): d/dx[log₁₀(x)] = 1/(x·ln(10))
⚡ Quick Reference Summary
Most Frequently Used Rules
| Rule/Function | Formula |
|---|---|
| Power Rule | d/dx[xⁿ] = nxⁿ⁻¹ |
| Product Rule | d/dx[uv] = u'v + uv' |
| Quotient Rule | d/dx[u/v] = (u'v - uv')/v² |
| Chain Rule | d/dx[f(g(x))] = f'(g(x))·g'(x) |
| e | d/dx[eˣ] = eˣ |
| ln(x) | d/dx[ln(x)] = 1/x |
| sin(x) | d/dx[sin(x)] = cos(x) |
| cos(x) | d/dx[cos(x)] = -sin(x) |
| tan(x) | d/dx[tan(x)] = sec²(x) |
| arctan(x) | d/dx[arctan(x)] = 1/(1 + x²) |
Focus on mastering these 10 rules first. They cover 95% of calculus problems you'll encounter. Once these are automatic, the rest becomes much easier!
🚀 Practice Your Derivatives!
Use our free derivative calculators to check your work and see step-by-step solutions!
Try Our Calculators →📄 Using This Cheat Sheet
- Print it out: Click the print button at the top or bottom of the page for a clean, printer-friendly version
- Keep it handy: Put it in your notebook, on your desk, or save it on your phone
- Practice daily: Review one section each day until these rules become automatic
- Use active recall: Cover the formulas and try to remember them before checking
- Make flashcards: Create cards for the rules you find most challenging
- Teach someone: Explaining these rules to others helps solidify your understanding
Study Schedule Recommendation
Week 1: Basic Rules
Focus on constant, power, constant multiple, and sum/difference rules. Practice 20 problems daily.
Week 2: Product, Quotient, Chain
Master these three essential rules. They're trickier but appear everywhere. Practice 15 problems daily.
Week 3: Trigonometric Functions
Memorize all six trig derivatives and their patterns. Use mnemonics. Practice 15 problems daily.
Week 4: Exponential & Logarithmic
Focus on e and ln, then move to other bases. Practice 15 problems daily.
Week 5: Mixed Practice
Combine all rules in complex problems. Practice 20 mixed problems daily.
📌 Key Takeaways
- ✅ Power Rule (xⁿ → nxⁿ⁻¹) is the foundation - master it first
- ✅ Product Rule: derivative of first × second + first × derivative of second
- ✅ Quotient Rule: (low d-high minus high d-low) / (low squared)
- ✅ Chain Rule: derivative of outside × derivative of inside
- ✅ eˣ is special: it's its own derivative!
- ✅ ln(x) → 1/x: one of the most useful derivatives
- ✅ sin → cos, cos → -sin: remember the negative for cos
- ✅ Co-functions have negatives: cos, cot, csc derivatives are negative
- ✅ Always check for chain rule: when you see composition, multiply by inner derivative
- ✅ Practice makes perfect: do problems daily until rules become automatic!
Don't just memorize formulas - understand WHY they work. When you understand the logic behind each rule, you'll remember them naturally and make fewer mistakes. Use our calculators to check your work and see step-by-step solutions when you're stuck. With consistent practice, these rules will become second nature!