CalcRef
Derivative Rules Explained: Power, Product, Quotient, Chain
Four rules do most of the work in differential calculus. Once the power, product, quotient, and chain rules are automatic — and you can tell at a glance which one a function calls for — differentiation stops being a puzzle and becomes bookkeeping. This guide explains each rule, the logic behind it, and the recognition skill that textbooks rarely teach directly: reading a function's structure before touching a pencil.
The power rule: the workhorse
For any real exponent n: d/dx xⁿ = n·xⁿ⁻¹. Bring the exponent down, reduce it by one. Combined with linearity — derivatives pass through sums and constant multiples — the power rule alone differentiates every polynomial.
Its real power appears when you rewrite first: √x = x^(1/2) and 1/x³ = x⁻³ both surrender to the same rule. Many "hard" derivatives are one algebraic rewrite away from being easy, and rewriting before differentiating is the single most underused habit in Calc I.
The product rule: not what intuition says
(fg)′ = f′g + fg′. The tempting wrong answer — that the derivative of a product is the product of derivatives — fails immediately: try f = g = x, where the true derivative of x² is 2x but the naive guess gives 1. The correct rule has a symmetric rhythm worth verbalizing while you work: derivative of the first times the second, plus the first times the derivative of the second. The symmetry also means you can't apply it "in the wrong order," which makes it one of the safer rules under exam pressure.
The quotient rule: keep the order straight
(f/g)′ = (f′g − fg′) / g², valid where g(x) ≠ 0. Unlike the product rule, order matters — the minus sign makes the numerator antisymmetric, and swapping the terms flips the sign of the whole answer. The classic mnemonic keeps the order straight: "low d-high minus high d-low, over low-low." Two practical notes:
- When the denominator is a single power, skipping the quotient rule is often cleaner: rewrite f(x)/x² as f(x)·x⁻² and use the product rule.
- The derivatives of tan, cot, sec, and csc all come from the quotient rule applied to sin and cos — deriving them once is the best way to stop misremembering their signs.
The chain rule: the one that runs everything
For a composition: d/dx f(g(x)) = f′(g(x)) · g′(x). Differentiate the outer function, evaluated at the inner function untouched, then multiply by the derivative of the inner function. The most common error is differentiating the outside and forgetting the inner factor — writing cos(x²) instead of 2x·cos(x²).
The chain rule matters beyond composite functions on homework: it is the engine inside implicit differentiation, related rates, and u-substitution (which is the chain rule read backwards). If one rule deserves overlearning, it is this one.
Recognition: which rule does this function want?
Ask what the function's last operation is — the outermost thing done to x:
- Last operation is a power of x (after rewriting roots and reciprocals) → power rule.
- Two x-containing expressions multiplied → product rule.
- One x-containing expression divided by another → quotient rule (or rewrite and use the product rule).
- A function wrapped around another (sin of, e-to-the, ln of, a power of a bracket) → chain rule.
Real problems nest these: x²·sin(3x) is a product whose second factor needs the chain rule. Work outside-in, one layer at a time, and every nested derivative decomposes into the four basics.
The base derivatives you still have to know
The four rules combine building blocks; the blocks themselves are memorized facts:
| Function | Derivative | Watch out for |
|---|---|---|
| sin x | cos x | no sign change |
| cos x | −sin x | the minus sign |
| tan x | sec²x | from quotient rule on sin/cos |
| eˣ | eˣ | its own derivative |
| aˣ | aˣ ln a | the ln a factor |
| ln x | 1/x | x > 0 |
| arcsin x | 1/√(1−x²) | |x| < 1 |
| arctan x | 1/(1+x²) | no square root |
Notice that every entry carries a condition or a trap alongside the formula — which is why memorizing a derivative without its domain and its classic error is only half the job. Our memorization guide covers how to drill these so the signs and conditions come along for free, and the AP formula review shows where each family appears on the exam.
How CalcRef helps
CalcRef's Derivative Rules reference table collects the power, trig, exponential, logarithmic, and inverse-trig derivatives in one screen-ready table, and its Derivatives topic — one of eight covering the full sequence — presents each formula with LaTeX-rendered notation, a plain-language description, conditions for use, and variable definitions, so the "watch out for" column above is built into the reference itself. When you're ready to make the rules automatic, flashcard quiz rounds of 10 cards drill the Derivatives topic (or mix all topics), with per-topic stats and quiz history tracking your progress. Free, offline, on iPhone, iPad, and Android.
Quick answers
Chain rule vs. product rule — how do I tell?
Multiplication between two x-expressions means product rule; wrapping (one function applied to another's output) means chain rule. x·sin x is a product; sin(x²) is a composition. Many functions need both — decide layer by layer, outside in.
Do I ever need the quotient rule?
Strictly, no — any quotient can be rewritten as a product with a negative or reciprocal power. Practically, yes: for messy denominators the quotient rule is faster and less error-prone than rewriting. Learn it, but remember the rewrite escape hatch.