AP Calculus derivative rules

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

AP Calculus exams do not give you a formula sheet, so you need these derivative rules by heart. Here they are with short examples.

Basic rules

• Constant: d/dx(c) = 0
• Power: d/dx(xⁿ) = nxⁿ⁻¹. Example: d/dx(x³) = 3x²
• Constant multiple and sum: differentiate term by term.

Product, quotient and chain rules

• Product: (uv)' = u'v + uv'. Example: d/dx(x² sin x) = 2x sin x + x² cos x
• Quotient: (u/v)' = (u'v − uv')/v²
• Chain: d/dx f(g(x)) = f'(g(x))·g'(x). Example: d/dx(sin 3x) = 3 cos 3x

Trigonometric functions

• d/dx(sin x) = cos x and d/dx(cos x) = −sin x
• d/dx(tan x) = sec²x and d/dx(cot x) = −csc²x
• d/dx(sec x) = sec x tan x and d/dx(csc x) = −csc x cot x

Exponential and logarithmic functions

• d/dx(e^x) = e^x and d/dx(a^x) = a^x ln a
• d/dx(ln x) = 1/x and d/dx(log_ax) = 1/(x ln a)

Inverse trigonometric functions

• d/dx(arcsin x) = 1/√(1 − x²)
• d/dx(arccos x) = −1/√(1 − x²)
• d/dx(arctan x) = 1/(1 + x²)

Other techniques

• Implicit differentiation: differentiate both sides with respect to x and attach dy/dx to every y term.
• Inverse functions: if g is the inverse of f, then g'(x) = 1/f'(g(x)).

See all the AP Calculus formulas to memorise.

Worked examples for every rule

Power rule

Differentiate 5x⁴ − 3x² + 7.

• 20x³ − 6x.

Product rule

Differentiate x²e^x.

• 2x e^x + x²e^x = e^x(x² + 2x).

Quotient rule

Differentiate (sin x)/x.

• (x cos x − sin x)/x².

Chain rule with a root

Differentiate √(1 + x²).

• Write it as (1 + x²)^1/2: ½(1 + x²)^−1/2 × 2x = x/√(1 + x²).

Chain rule with exponentials and logs

Differentiate e^x² and ln(3x² + 1).

• d/dx e^x² = 2x e^x².
• d/dx ln(3x² + 1) = 6x/(3x² + 1).

Implicit differentiation

Find dy/dx for x² + xy + y² = 7.

• 2x + (y + x y') + 2y y' = 0.
• y'(x + 2y) = −(2x + y), so y' = −(2x + y)/(x + 2y).

Derivative of an inverse

f(x) = x³ + x and g is its inverse. Find g'(2).

• f(1) = 2, so g(2) = 1. f'(x) = 3x² + 1, so f'(1) = 4.
• g'(2) = 1/f'(1) = 1/4.

Second derivatives

Find d²y/dx² for y = sin(2x).

• y' = 2cos(2x), so y'' = −4 sin(2x).

A decision guide

1. Is it a sum of simple terms? Differentiate term by term.
2. Is it a product of two functions? Product rule.
3. Is it a fraction? Quotient rule, or rewrite with a negative power.
4. Is there a function inside a function? Chain rule, every time.
5. Are x and y mixed in one equation? Implicit differentiation.

Practice questions

Try each one before opening the answer.

Key takeaways

• Learn the power, product, quotient and chain rules until they are automatic.
• Memorise the trig, exponential, log and inverse trig derivatives.
• Apply the chain rule whenever there is an inner function.
• Use implicit differentiation for equations mixing x and y.
• Practise mixed questions where you must choose the rule.

Why the rules work

Every derivative rule comes from the same definition: the derivative is the limit of the slope of a secant line as the two points get closer together. You do not need to prove each rule in the exam, but understanding where they come from makes them easier to remember and harder to misuse.

The power rule

For x², the slope between x and x + h is ((x + h)² − x²)/h = 2x + h, which approaches 2x as h shrinks. The same pattern for xⁿ gives nxⁿ⁻¹. This is why the exponent "comes down" and decreases by one.

The product rule

Think of a rectangle with sides u and v. If both sides grow slightly, the area grows by about u·(change in v) + v·(change in u). That is exactly the product rule. It also explains why the derivative of a product is not the product of the derivatives.

The chain rule

If y changes three times as fast as u, and u changes twice as fast as x, then y changes six times as fast as x. Rates multiply through a chain of functions. That is all the chain rule says: dy/dx = (dy/du)(du/dx).

How derivative rules appear in the exam

• Direct questions: differentiate a function, often combining two or three rules.
• Tables: you are given values of f, g, f' and g' at certain points and asked for the derivative of f(g(x)) or f(x)g(x) at a point.
• Graphs: you read slopes from a graph and combine them with a rule.
• Applications: derivatives feed into tangent lines, related rates, optimisation and motion questions.

Example: derivatives from a table

Suppose f(2) = 3, f'(2) = 4, g(1) = 2 and g'(1) = 5. Find the derivative of f(g(x)) at x = 1.

• By the chain rule, d/dx f(g(x)) = f'(g(x))·g'(x).
• At x = 1: f'(g(1))·g'(1) = f'(2) × 5 = 4 × 5 = 20.

More practice

Try each one before opening the answer.

Frequently asked questions

Do I need to know the derivative of every inverse trig function?

Know arcsin, arccos and arctan. They appear regularly, especially arctan.

Should I simplify my derivatives?

Simplify enough to use the result. In free-response answers, an unsimplified but correct derivative usually earns credit, but arithmetic slips while simplifying can cost marks.

What is the most common derivative mistake?

Forgetting the chain rule on an inner function, such as writing d/dx(sin 3x) = cos 3x instead of 3cos 3x.

Study tip

Write each rule on a flashcard with one example on the back. Shuffle them daily for two weeks, then switch to mixed questions where you must choose the rule yourself.

Where to go next

Once the rules feel automatic, use them in the applications that carry the most marks: related rates, motion and curve analysis.

• Related rates, step by step
• AP Calculus AB units, explained
• AP Calculus AB review and study plan

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