🎓 Exam-Ready Edition · Updated March 23, 2026
Derivative Rules Cheat Sheet: Complete Printable Guide for Calculus Exams [2026]
Every derivative rule you need on exam day — Power, Chain, Product, Quotient, all Trig derivatives, Inverse Trig, Exponential, Logarithmic, and Implicit Differentiation. Each rule includes the formula, a quick example, and expert memory tips. 100% free. One-page printable.
⚡ Jump to a Rule:
Foundational Rules
1
Constant Rule
Formula
d/dx [c] = 0
Quick Example
f(x) = 7
f'(x) =
0
Constants have no rate of change.
2
Power Rule
Formula
d/dx [xn] = n · xn−1
Quick Example
f(x) = x5
f'(x) =
5x4
Works for any real exponent n.
3
Constant Multiple Rule
Formula
d/dx [c·f(x)] = c · f'(x)
Quick Example
f(x) = 4x3
f'(x) =
12x2
Pull the constant out; differentiate normally.
4
Sum / Difference Rule
Formula
d/dx [f ± g] = f'(x) ± g'(x)
Quick Example
f(x) = x3 − 5x
f'(x) =
3x2 − 5
Differentiate term by term.
Combination Rules
5
Product Rule
Formula
d/dx [f·g] = f'·g + f·g'
Quick Example
f(x) = x2·sin x
f'(x) =
2x·sin x + x2·cos x
Use when two functions are multiplied.
6
Quotient Rule
Formula
d/dx [f/g] = (g·f' − f·g') / g2
Quick Example
f(x) = x2 / (x+1)
f'(x) =
(x2+2x) / (x+1)2
Use when one function is divided by another.
7
Chain Rule ⭐ Most Used
Formula
d/dx [f(g(x))] = f'(g(x)) · g'(x)
Quick Example
f(x) = sin(3x2)
f'(x) =
cos(3x2) · 6x
Use for composite functions — function inside a function.
Trigonometric Derivatives
8
All 6 Trig Derivatives
| Function | Derivative Formula | Quick Example |
|---|---|---|
| sin x | cos x | d/dx[sin(2x)] = 2cos(2x) |
| cos x | −sin x | d/dx[cos(x²)] = −2x·sin(x²) |
| tan x | sec²x | d/dx[tan(3x)] = 3sec²(3x) |
| csc x | −csc x · cot x | d/dx[csc(x)] = −csc x cot x |
| sec x | sec x · tan x | d/dx[sec(x)] = sec x tan x |
| cot x | −csc²x | d/dx[cot(5x)] = −5csc²(5x) |
💡 Pattern: sin→cos→−sin→−cos (cycle). The "co" functions (cos, csc, cot) all get a negative sign.
Inverse Trigonometric Derivatives
9
Inverse Trig Derivatives
| Function | Derivative Formula | Quick Example |
|---|---|---|
| arcsin x | 1 / √(1−x²) | d/dx[arcsin(2x)] = 2/√(1−4x²) |
| arccos x | −1 / √(1−x²) | d/dx[arccos(x)] = −1/√(1−x²) |
| arctan x | 1 / (1+x²) | d/dx[arctan(3x)] = 3/(1+9x²) |
| arccsc x | −1 / (|x|√(x²−1)) | Rare — note the absolute value |
| arcsec x | 1 / (|x|√(x²−1)) | Common in advanced calc |
| arccot x | −1 / (1+x²) | Negative of arctan derivative |
💡 For exams: memorize arcsin, arccos, arctan — the others are rarely tested.
Exponential & Logarithmic Derivatives
10
Natural Exponential
Formula
d/dx [ex] = ex
Quick Example
f(x) = e3x
f'(x) =
3e3x
ex is the only function that is its own derivative!
11
General Exponential (aˣ)
Formula
d/dx [ax] = ax · ln(a)
Quick Example
f(x) = 2x
f'(x) =
2x · ln(2)
Multiply by ln(a); when a=e, ln(e)=1.
12
Natural Log (ln x)
Formula
d/dx [ln x] = 1/x
Quick Example
f(x) = ln(x²+1)
f'(x) =
2x / (x²+1)
Chain rule applies: d/dx[ln(u)] = u'/u.
13
Log Base a (loga x)
Formula
d/dx [loga x] = 1 / (x · ln a)
Quick Example
f(x) = log2(x)
f'(x) =
1 / (x · ln 2)
Convert via change of base: loga(x) = ln(x)/ln(a).
Implicit Differentiation
14
Implicit Differentiation — Step-by-Step
Use when y cannot be isolated explicitly, e.g. x² + y² = 25 (circle).
1
Differentiate both sides w.r.t. x
Apply d/dx to every term on both sides of the equation.
2
Apply Chain Rule to y terms
Whenever you differentiate a y term, multiply by dy/dx: d/dx[y²] = 2y·(dy/dx)
3
Collect all dy/dx terms on one side
Move terms with dy/dx to the left, everything else to the right.
4
Factor out dy/dx and solve
Divide both sides to isolate dy/dx.
Classic Example: x² + y² = 25
Step 1: 2x + 2y·(dy/dx) = 0
Step 2: 2y·(dy/dx) = −2x
Step 3: dy/dx = −x/y
Step 2: 2y·(dy/dx) = −2x
Step 3: dy/dx = −x/y
Another Example: x³ + y³ = 6xy
3x² + 3y²·(dy/dx) = 6y + 6x·(dy/dx)
dy/dx·(3y² − 6x) = 6y − 3x²
dy/dx = (6y − 3x²) / (3y² − 6x)
💡 Memory Tips for the Hardest Rules
Quotient Rule — "Low D-High, High D-Low"
For d/dx[f/g], say out loud: "Low times derivative of High, minus High times derivative of Low, over Low squared."
Lo·dHi − Hi·dLo / Lo²Chain Rule — "Outside–Inside"
Differentiate the outer function first (keep inner unchanged), then multiply by the derivative of the inner function.
d/dx[outer(inner)] = outer'(inner) × inner'Product Rule — "First · D-Second + Second · D-First"
Keep the first function, differentiate the second, then add the second function times the derivative of the first.
f·g' + g·f' (never subtract!)Trig Derivatives — "co = negative"
All "co" trig functions (cos, csc, cot) get a negative sign when differentiated. Only the positive ones stay positive.
sin→+cos tan→+sec² sec→+sec·tanInverse Trig — "The Big Three"
For exams, focus on arcsin (1/√(1−x²)), arccos (−1/√(1−x²)), and arctan (1/(1+x²)). arccos = −arcsin.
arctan → 1/(1+x²) ← most tested!Implicit Differentiation — "Tag every y"
Every time you see a y term, differentiate it normally, then immediately "tag" it with ×(dy/dx). Treat x terms normally.
d/dx[y³] = 3y²·(dy/dx) ← don't forget!✅ Exam-Day Strategy: 8 Rules to Never Forget
Identify first Look at the function structure before picking a rule.
Chain Rule is everywhere Any f(g(x)) needs it — even sin(2x).
Quotient → Product tip Rewrite f/g as f·g⁻¹ to use Product Rule instead.
Check signs on trig Forgetting the minus on cos, csc, cot is the #1 error.
Simplify before differentiating Algebra first saves time and errors.
Use ln for tricky exponentials Logarithmic differentiation helps with xx or [f(x)]g(x).
Tag every y in implicit Every y term gets a dy/dx factor — don't skip one.
Factor your final answer Professors award more marks for simplified derivatives.
❓ Frequently Asked Questions
What derivative rules do I need to know for a calculus exam?
▼
For most calculus exams (AP Calculus, Calc 1 & 2), you need: Power Rule, Product Rule, Quotient Rule, Chain Rule, all 6 Trig derivatives, the three key Inverse Trig derivatives (arcsin, arccos, arctan), Exponential (eˣ and aˣ), Logarithmic (ln x and loga x), and Implicit Differentiation. This cheat sheet covers all of them.
Can I print this derivative cheat sheet?
▼
Yes! Click the "Print This Page" button at the top of the page. The page uses specially optimized print styles — all navigation, backgrounds, and non-essential sections are hidden during printing so the rules and memory tips compress cleanly onto one A4 or Letter-size sheet.
What is the most important derivative rule for exams?
▼
The Chain Rule is the most critical. It appears in over 80% of differentiation problems because most functions you encounter — sin(2x), ex², ln(3x+1) — are composite functions. Master the Chain Rule above all others.
How do I remember the Quotient Rule without forgetting the minus sign?
▼
Use the "Low D-High minus High D-Low over Low-Low" mnemonic: d/dx[f/g] = (g·f' − f·g') / g². The key is that it's minus (not plus like the Product Rule), and the denominator is g squared. Say the mnemonic aloud while writing it out — repetition builds muscle memory.
What's the difference between Product Rule and Chain Rule?
▼
Product Rule: Use when two functions are multiplied together, like x²·sin x — both are "standalone" functions being multiplied. Chain Rule: Use when one function is inside another, like sin(x²) — here x² is the input to sin. Check: is there a function inside another function? If yes, Chain Rule. Are two separate functions multiplied? If yes, Product Rule. Often, you'll need both at once.
When do I use implicit differentiation?
▼
Use implicit differentiation when an equation has both x and y mixed together and you cannot easily solve for y explicitly. Classic examples: x² + y² = r² (circle), x³ + y³ = 6xy, or any conic section. The technique: differentiate both sides with respect to x, apply Chain Rule to every y term (which gives a dy/dx factor), then solve for dy/dx.