What is the most important derivative rule to know?

The chain rule is the most important derivative rule for AP Calculus because nearly every real exam problem involves a composite function. If you can reliably identify the outer and inner functions and multiply their derivatives, you can handle most differentiation problems.

How do I know when to use the chain rule?

Use the chain rule any time one function is nested inside another — for example, sin(x²), (3x+1)⁴, or e^(2x). If you can identify an inner function u = g(x) that is not simply x, the chain rule is required. It is the most commonly missed step in AP Calculus differentiation.

Do I need to memorize all trig derivatives for AP Calculus?

For AP Calculus AB, you should memorize the derivatives of sin, cos, and tan at minimum. The derivatives of sec, csc, and cot appear less frequently but are still fair game. All six are on the AP Calculus formula sheet, so you can reference them during the exam — but recognizing them on sight saves time.

What is the difference between the product rule and the chain rule?

Use the product rule when two functions are multiplied together: d/dx[f·g] = f'g + fg'. Use the chain rule when one function is composed inside another: d/dx[f(g(x))] = f'(g(x))·g'(x). Many problems require both rules applied together — for example, differentiating x²·sin(x³) uses the product rule on the outside and the chain rule for sin(x³).

Can I use a derivative calculator to check my work?

Yes. The MathInSite derivative calculator differentiates polynomials term by term using the power rule and shows each step of the working. Use it to verify your answer after applying the rules manually — this is the most effective way to catch errors in the chain rule or product rule application.

Derivative Rules Cheat Sheet: Every Formula + Worked Example

This derivative rules cheat sheet covers every differentiation rule you need for AP Calculus AB, AP Calculus BC, and Calculus 1 — one rule per section, formula and notation, a fully worked example, and the common mistake to avoid. Use it as a study reference or print it for exam day.

Every rule in this guide also links to MathInSite's derivative calculator, part of MathInSite's suite of free math calculators, so you can verify any example with step-by-step working. No login required.

Formulas verified against College Board AP Calculus AB/BC course descriptions and standard calculus textbooks. Last reviewed September 2026.

Jump to a rule: Notation · Constant · Power · Constant Multiple · Sum & Difference · Product · Quotient · Chain · Trig · Inverse Trig · Exponential · Logarithmic

Notation Guide: Two Ways to Write a Derivative

Both notations mean the same thing and appear interchangeably in textbooks and on AP exams.

Notation	Meaning	Common context
f'(x)	The derivative of f at x	Lagrange/prime notation — textbooks, function problems
dy/dx	The derivative of y with respect to x	Leibniz notation — calculus course work, related rates
d/dx [f(x)]	Apply the derivative operator to f(x)	Leibniz — emphasizes the operation being performed
y'	The derivative of y	Shorthand when the variable is clear from context

Constant Rule

d/dx [c] = 0

The derivative of any constant is zero — a constant does not change, so its rate of change is zero.

Examples: d/dx [7] = 0 · · d/dx [−3] = 0 · · d/dx [π] = 0

Common mistake: Confusing d/dx[c] = 0 with d/dx[c^x], which is an exponential function and is not zero. The constant rule only applies when c is the entire expression, not the base of an exponent.

Power Rule

d/dx [xⁿ] = n · xⁿ⁻¹ (for any real n)

Multiply by the exponent, then decrease the exponent by 1.

Example 1 (integer exponent):

d/dx [x⁵] = 5x⁴

Example 2 (fractional exponent):

d/dx [x^1/2] = (1/2)x^−1/2 = 1/(2√x)

Common mistake: Forgetting to subtract 1 from the exponent. d/dx[x⁵] = 5x⁵ is wrong — the exponent must become 5−1 = 4. Also, do not apply the power rule to e^x; that is the exponential rule.

Try the derivative calculator to verify power rule problems with a step-by-step breakdown.

Constant Multiple Rule

d/dx [c · f(x)] = c · f'(x)

A constant factor pulls out of the derivative. Differentiate the function, leave the constant alone.

Example: d/dx [5x³] = 5 · 3x² = 15x²

Sum and Difference Rule

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Differentiation is a linear operation — you can split it across addition and subtraction and differentiate term by term.

Example: d/dx [x³ + 4x² − 7] = 3x² + 8x − 0 = 3x² + 8x

Product Rule

d/dx [f(x) · g(x)] = f'(x) · g(x) + f(x) · g'(x)

Memory device: "derivative of first times second, plus first times derivative of second."

Worked example: d/dx [x² · sin x]

Identify: f = x², g = sin x
Differentiate: f' = 2x, g' = cos x
Apply: f'g + fg' = 2x · sin x + x² · cos x
Answer: 2x sin x + x² cos x

Common mistake: d/dx[f · g] ≠ f' · g'. You cannot simply multiply the individual derivatives — you must use the product rule formula.

Quotient Rule

d/dx [f(x) / g(x)] = (f'(x)·g(x) − f(x)·g'(x)) / [g(x)]²

Mnemonic: "Low D-High minus High D-Low, over Low squared" — Lo dHi − Hi dLo / Lo²

Worked example: d/dx [sin x / x²]

Identify: f = sin x (High), g = x² (Low)
Differentiate: f' = cos x, g' = 2x
Apply: (x² · cos x − sin x · 2x) / x⁴
Simplify: (x² cos x − 2x sin x) / x⁴ = (x cos x − 2 sin x) / x³
Answer: (x cos x − 2 sin x) / x³

Common mistake: Reversing the subtraction order in the numerator. It is always f'g − fg', never fg' − f'g. Getting this backwards flips the sign of the answer.

Chain Rule

d/dx [f(g(x))] = f'(g(x)) · g'(x)

Method: Differentiate the outer function (leaving the inner function intact), then multiply by the derivative of the inner function. The chain rule is required in over 80% of AP Calculus differentiation problems.

Example 1 (basic): d/dx [(3x + 1)⁴]

Outer function: u⁴ → 4u³. Inner function: 3x + 1 → 3
Apply: 4(3x + 1)³ · 3
Answer: 12(3x + 1)³

Example 2 (AP-level composite): d/dx [sin(x²)]

Outer function: sin(u) → cos(u). Inner function: x² → 2x
Apply: cos(x²) · 2x
Answer: 2x cos(x²)

Common mistake: Forgetting to multiply by the inner derivative — the most common AP exam error. d/dx[sin(x²)] ≠ cos(x²). The chain rule always requires that final multiplication by g'(x).

Verify your chain rule answers with the derivative calculator — it shows every step.

Trigonometric Derivatives

Function	Derivative
d/dx [sin x]	cos x
d/dx [cos x]	−sin x
d/dx [tan x]	sec²x
d/dx [cot x]	−csc²x
d/dx [sec x]	sec x · tan x
d/dx [csc x]	−csc x · cot x

Memory tip: The derivatives of the "co-functions" (cos, cot, csc) all carry a negative sign. Sin and cos cycle into each other; the pattern repeats.

Chain rule applied to trig: d/dx [tan(2x)] = sec²(2x) · 2 = 2sec²(2x)

Always check whether the trig function has an inner function. If yes, apply the chain rule.

Explore trig function values with the trig functions calculator.

Inverse Trigonometric Derivatives

Function	Derivative	Domain
d/dx [arcsin x]	1 / √(1 − x²)	\|x\| < 1
d/dx [arccos x]	−1 / √(1 − x²)	\|x\| < 1
d/dx [arctan x]	1 / (1 + x²)	all real x

BC / Calculus 2 completeness: arccot x → −1/(1+x²); arcsec x → 1/(|x|√(x²−1)); arccsc x → −1/(|x|√(x²−1)).

Worked example (chain rule applied): d/dx [arctan(3x)]

Outer: arctan(u) → 1/(1+u²). Inner: 3x → 3
Apply: 1/(1+(3x)²) · 3 = 3/(1+9x²)
Answer: 3/(1 + 9x²)

Exponential Derivatives

d/dx [e^x] = e^x

e^x is the only function equal to its own derivative — this is what makes e the "natural" base for calculus.

d/dx [a^x] = a^x · ln(a), where a > 0 and a ≠ 1

Example 1 (chain rule): d/dx [e^3x] = e^3x · 3 = 3e^3x

Example 2 (general base): d/dx [2^x] = 2^x · ln 2

Logarithmic Derivatives

d/dx [ln x] = 1/x, x > 0

d/dx [log_a x] = 1/(x · ln a), x > 0

Example 1 (chain rule applied): d/dx [ln(x² + 1)] = (2x)/(x² + 1)

Example 2 (log base change): d/dx [log₂ x] = 1/(x · ln 2)

These logarithmic derivative rules connect directly to the change-of-base formula. Work with logarithms step by step using the logarithm calculator.

Which Derivative Rule Do I Use? A Decision Guide

Students who know the derivative rules often freeze on rule selection. Use this checklist in order:

Is the function just a constant? → Constant Rule: d/dx[c] = 0
Is it x raised to a power? → Power Rule: d/dx[xⁿ] = nxⁿ⁻¹
Is it a constant times a function? → Constant Multiple Rule: d/dx[cf] = cf'
Is it a sum or difference of functions? → Sum/Difference Rule: differentiate term by term
Is it two functions multiplied together? → Product Rule: f'g + fg'
Is it one function divided by another? → Quotient Rule: (f'g − fg') / g²
Is one function nested inside another? → Chain Rule — always check this last, and apply it in combination with the other rules
Is it a trig, inverse trig, exponential, or logarithmic function? → Use the derivative rules tables above

Key insight: Many real differentiation rules problems require more than one rule at once. For example, d/dx[x² sin(x³)] requires the product rule on the outside and the chain rule for sin(x³). Always check for composite functions after identifying the outermost structure.

Common Derivative Mistakes to Avoid

Applying power rule to e^x: d/dx[e^x] ≠ x·e^x−1. Correct: d/dx[e^x] = e^x.
Forgetting the chain rule: d/dx[sin(x²)] ≠ cos(x²). Correct: 2x·cos(x²).
Reversing quotient rule subtraction: Numerator must be f'g − fg', not fg' − f'g.
Treating product like addition: d/dx[f(x)·g(x)] ≠ f'(x)·g'(x). Use the product rule.
Dropping the inner derivative (chain): Always multiply by g'(x). This is the single most common AP Calculus differentiation error.

Printable Derivative Rules Reference Card

Rule	Formula	Quick example
Constant	d/dx[c] = 0	d/dx[5] = 0
Power	d/dx[xⁿ] = nxⁿ⁻¹	d/dx[x⁴] = 4x³
Constant Multiple	d/dx[cf] = cf'	d/dx[3x²] = 6x
Sum/Difference	d/dx[f±g] = f'±g'	d/dx[x²+x] = 2x+1
Product	d/dx[fg] = f'g+fg'	d/dx[x·sin x] = sin x+x cos x
Quotient	d/dx[f/g] = (f'g−fg')/g²	Lo dHi − Hi dLo / Lo²
Chain	d/dx[f(g(x))] = f'(g(x))·g'(x)	d/dx[(x²+1)³] = 3(x²+1)²·2x
sin x	cos x	d/dx[sin(3x)] = 3cos(3x)
cos x	−sin x	d/dx[cos(2x)] = −2sin(2x)
tan x	sec²x	d/dx[tan x] = sec²x
e^x	e^x	d/dx[e^5x] = 5e^5x
ln x	1/x	d/dx[ln(x²)] = 2x/x² = 2/x
arctan x	1/(1+x²)	d/dx[arctan(2x)] = 2/(1+4x²)

Use this derivative rules cheat sheet as a one-page printable reference for your AP Calculus exam. Press Cmd+P (Mac) or Ctrl+P (Windows) to print or save as PDF. Navigation and decorative elements are suppressed in print view.

Frequently Asked Questions

What is the most important derivative rule to know?

The chain rule is the most important derivative rule for AP Calculus because nearly every real exam problem involves a composite function. If you can reliably identify the outer and inner functions and multiply their derivatives, you can handle most differentiation problems.
How do I know when to use the chain rule?

Use the chain rule any time one function is nested inside another — for example, sin(x²), (3x+1)⁴, or e^2x. If you can identify an inner function u = g(x) that is not simply x, the chain rule is required. It is the most commonly missed step in AP Calculus differentiation.
Do I need to memorize all trig derivatives for AP Calculus?

For AP Calculus AB, memorize the derivatives of sin, cos, and tan at minimum. The derivatives of sec, csc, and cot are fair game and the College Board provides them on the formula sheet, but recognizing them on sight saves time during the exam.
What is the difference between the product rule and the chain rule?

Use the product rule when two functions are multiplied: d/dx[f·g] = f'g + fg'. Use the chain rule when one function is composed inside another: d/dx[f(g(x))] = f'(g(x))·g'(x). Many problems require both — for example, differentiating x²·sin(x³) uses the product rule on the outside and the chain rule for sin(x³).
Can I use a derivative calculator to check my work?

Yes. The MathInSite derivative calculator differentiates polynomials using the power rule and shows each step. Use it to verify your answer after applying the rules manually — this is the most effective way to catch chain rule or product rule errors.

Ready to practice? Use the derivative calculator to verify any problem from this guide. When you're ready for the next step, the integral calculator covers antiderivatives — the inverse operation. Browse all 27 free math calculators at MathInSite.

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