Every derivative you will ever compute by hand is one of four moves — power, product, quotient, chain — plus the housekeeping rules for constants and sums. This guide states each rule once, applies it once to a clean example, and names the mistake that rule is famous for. If you arrived looking for a quotient rule derivative calculator, the honest position of this site is in the open: the Derivative Calculator differentiates term by term, so products, quotients and nestings are worked here, by hand, where the working can be shown. Rules first, machinery second, traps labelled.

The power rule: bring it down, take one away

The power rule runs d/dx [x^n] = n·x^(n−1): the old exponent becomes a multiplier and the new exponent is one smaller. Applied to x^7: the 7 comes down and the power drops to 6, giving 7x^6. At x = 1 the slope is exactly 7. The rule does not care what n is. Negative powers obey it identically — x^-3 differentiates to −3x^-4, the multiplier −3 arriving with its sign attached — and fractional powers turn it into the root rule: x^0.5 (the square root, typed the way the calculator expects) differentiates to 0.5x^-0.5, which is 1/(2√x) in root notation. At x = 4 that slope is 0.25: the root curve is still climbing there, but flattening, exactly as its graph promises.

The mistake this rule is famous for is applying it where the variable is not in the base. x^7 has the variable downstairs and earns 7x^6; 7^x has the variable upstairs, in the exponent, and the power rule simply does not live there — 7^x differentiates to 7^x · ln 7, a different family entirely (the companion article Common Derivatives derives the whole a^x row). One second spent asking “where is the x?” prevents the most expensive error in the topic.

Coefficients and sums: the housekeeping rules

Two rules do the quiet work inside every longer problem. The constant-multiple rule says a coefficient is a passenger: d/dx [c·f(x)] = c·f′(x). Applied to 5x^3, the x^3 part differentiates to 3x^2 and the 5 rides along: 15x^2, slope 60 at x = 2. The sum rule says a sum differentiates term by term: d/dx [f ± g] = f′ ± g′. Applied to 4x^3 − 2x^2 + x − 7, each term takes its own rule — power, power, power, constant-to-zero — and the results assemble into 12x^2 − 4x + 1, with slope 9 at x = 1. That worked sum is not an arbitrary example: it is the largest function the calculator at the top of the hub page was checked against for this article, all four of its terms, and its step display names the rule on every line. When a long derivative goes wrong, it goes wrong one term at a time — which is why the audit is also done one term at a time.

The product rule: each factor takes a turn

For a product f·g, the rule is d/dx [f·g] = f′g + fg′: differentiate the first factor while the second watches, then the second while the first watches, and add. Applied to (2x + 1)(x^2 − 3): the derivative of the first factor is 2, of the second 2x, so the assembly is 2(x^2 − 3) + (2x + 1)(2x) = 2x^2 − 6 + 4x^2 + 2x = 6x^2 + 2x − 6. At x = 2 the slope is 22. The famous mistake is to differentiate both factors and multiply the results — 2 × 2x = 4x, slope 8 at x = 2 — which throws away the cross-terms where all the growth actually lives. A product grows because both factors are moving; a rule that only lets them move one at a time, and adds the two contributions, is the only shape that can be right. Sanity-check the size whenever you use it: the true product derivative at a point is almost never the product of the derivatives, and when the two disagree wildly, the product rule — not the shortcut — is the one to believe.

The quotient rule: top, bottom, and the order that matters

For a quotient f/g, the rule is d/dx [f/g] = (f′g − fg′)/g^2. Three things are load-bearing: the numerator is a difference, the first product is f′g (derivative of the top, times the bottom), and the denominator is the bottom squared, whatever the bottom was. Applied to x^2/(x + 2): f′ = 2x and g′ = 1, so the numerator is 2x(x + 2) − x^2 · 1 = x^2 + 4x, and the derivative is (x^2 + 4x)/(x + 2)^2, or factored, x(x + 4)/(x + 2)^2. At x = 1 the slope is 5/9 ≈ 0.5556; at x = 2 it is 0.75 exactly.

The famous mistake is running the numerator backwards — (fg′ − f′g) instead of (f′g − fg′) — which negates the entire answer: the same example at x = 1 comes out −5/9. The fraction x^2/(x + 2) is plainly climbing at x = 1 (it is heading from 1/3 up through 1 at x = 2), so a negative slope there convicts the order immediately. Two habits make the rule safe: recite it in the same words every time — “derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared” — and run the sign sanity check at one easy point before simplifying anything. The mnemonic many courses teach, “lo-dee-hi minus hi-dee-lo over lo-lo”, is the same sentence compressed; use whichever you cannot say backwards by accident.

Four rules, four worked examples, four traps. The sum and constant-multiple rules are woven through the examples rather than rowed here — they are the housekeeping, not the machinery.

The chain rule: outside, then the inside’s toll

For a nesting f(g(x)) — a function whose input is another function — the rule is d/dx [f(g(x))] = f′(g(x)) · g′(x): differentiate the outside as if the inside were a plain variable, keep the inside untouched, then multiply by the derivative of the inside. Applied to (3x − 1)^4: the outside is a fourth power, giving 4(3x − 1)^3; the inside 3x − 1 differentiates to 3; the toll is paid and the answer is 12(3x − 1)^3 — slope 96 at x = 1. A trigonometric nesting obeys the same toll booth: sin(x^2) differentiates its outside to cos(x^2) and pays the inside’s derivative 2x, giving 2x·cos(x^2), slope 2cos(1) ≈ 1.0806 at x = 1.

The famous mistake is skipping the toll — writing 4(3x − 1)^3 and stopping. At x = 1 that reports a slope of 32 against the true 96: wrong by exactly the inside derivative, a factor of 3. The error is invisible in the shape of the answer, which is what makes it dangerous; only a numeric probe at a point exposes it. Build the probe habit from the checking routine on the hub page: evaluate your derivative and a symmetric difference quotient at one convenient x, and refuse any answer they disagree on. Every rule in this article passes that test at the points quoted — the worked values were verified against independent computation before a single one was printed.

Constants inside the rules: passengers and vanishing acts

Two constant behaviours cause a steady drip of lost marks, and both are one line long. A constant standing alone differentiates to zero: d/dx [9] = 0, and the −7 in the worked sum above vanished for exactly this reason. A constant multiplying a function does not vanish — it is carried through the function’s own rule untouched. So d/dx [4sin(x)] = 4cos(x): the sine rule supplies cos(x), the 4 rides as a passenger, and at x = 0 the slope is 4. The errors come from swapping the two behaviours — killing a passenger as if it stood alone (4sin(x) → cos(x)), or promoting a loner into a factor (9 → 9 · 0 done “to be safe”, which at least lands on 0 by accident). Ask of every constant: alone, or attached? The answer decides whether it disappears or commutes.

Rules in combination: the usual case

Textbook examples isolate one rule at a time; real functions stack them, and the stack is handled outside-in, one layer per pass. Take sin(x^2) + x^7. The sum rule splits it first: a trig nesting plus a plain power. The power is immediate — 7x^6. The nesting takes the chain rule: outside sine becomes cosine with the inside preserved, times the inside derivative 2x, giving 2x·cos(x^2). Assemble: 2x·cos(x^2) + 7x^6, and at x = 1 the slope is 2cos(1) + 7 ≈ 1.0806 + 7 = 8.0806. Nothing new was invented at any step — the combination is only the triage below, applied recursively: read the outermost structure, apply its rule, and hand whatever remains to the next pass.

Which rule am I in? A thirty-second triage

Real problems rarely announce their rule. Read the outermost structure of the function — the last operation that would be performed if you evaluated it at a number. If the last act is adding or subtracting terms, you are in sum country: split and take each term separately. If the last act is multiplying two x-containing pieces, product rule. If it is dividing, quotient rule. If it is feeding the whole expression into a power, a root, a sine or an exponential, chain rule — and the inside is whatever is being fed in. Constants multiplying a single piece never change the triage; they ride through whichever rule the piece itself needs. Triage first, algebra second: most wrong derivatives are not failed algebra at all, but a correctly-executed rule that was never the rule the function was asking for.