Most derivative questions are not new problems — they are the same thirty functions wearing different coefficients. This is the reference for them: the derivative of each function students actually meet, with its one-line derivation rather than just its answer, because a derivation can be re-run on the thirty-first function and an answer cannot. Every entry below was computed and then verified independently before it was printed; where the hub Derivative Calculator accepts the function, its own output is the second witness.

The reciprocal family: 1/x and its multiples

Start with the most-searched derivative on the internet. What is the derivative of 1/x? Rewrite the fraction as a power — 1/x is x^-1 — and let the power rule run: the derivative of x^-1 is −x^-2, i.e. −1/x^2. The minus sign is the content: 1/x falls as x grows, and at x = 2 the derivative 1/x question resolves to a slope of −0.25. The whole family follows by the constant-multiple rule. The derivative of 2/x is −2/x^2; the derivative of 3/x is −3/x^2; the derivative of 4/x is −4/x^2; the derivative of 5/x is −5/x^2. One pattern, five searches: d/dx [c/x] = −c/x^2, the coefficient carried and the sign flipped. The derivative of a fraction with x underneath is never a quotient-rule problem unless the top also contains x — it is a negative power asking to be recognised, which is why the derivative of fraction questions cluster here rather than in the quotient article.

Two sign curiosities complete the family. The derivative of -1/x is +1/x^2: the leading minus flips twice — once by the power rule’s own negative, once by the coefficient — and the function −1/x, which climbs toward zero from below, correctly reports a positive slope (0.25 at x = 2). And the fraction rule extends one step further: the derivative of 1/(1-x) is 1/(1-x)^2, by the chain rule applied to (1 − x)^-1 — the inner −x contributes a −1 that cancels the power’s own minus sign, which is why this one comes out positive. At x = 0 its slope is exactly 1.

The power family, including the quiet ones

The derivative of x^3 is 3x^2 (slope 12 at x = 2), and the derivative of 2x^2 is 4x — coefficient doubled by the power, slope 12 at x = 3. The linear family is quieter: the derivative of 3x is 3, the derivative of x/2 is 1/2, and the derivative of x/3 is 1/3 — a line’s derivative is its own slope, read straight off. Root forms are powers in costume: the derivative of x^1/2 is (1/2)x^-1/2, i.e. 1/(2√x), slope 0.25 at x = 4. Then the two entries students refuse to believe: the derivative of 1 is 0 and the derivative of 3 is 0 — any constant, however it is dressed, has no x in it, changes nowhere, and differentiates to nothing. The derivative of y deserves its own sentence because it starts the implicit story: with respect to y itself, the derivative of y is 1; when y is a function of x, its derivative is written y′ and cannot be simplified until the relation tying y to x is known — the seed of implicit differentiation.

The a^x family: the base survives, times its log

For a constant base, d/dx [a^x] = a^x · ln(a): the function reproduces itself and pays a toll of the natural log of its base. The derivative of 3^x is 3^x · ln 3, with ln 3 ≈ 1.0986 — at x = 2 the slope is 9 × 1.0986 ≈ 9.8875. The derivative of 2^x is 2^x · ln 2 (ln 2 ≈ 0.6931; slope ≈ 5.5452 at x = 3). The derivative of 4^x is 4^x · ln 4 (ln 4 ≈ 1.3863; slope ≈ 5.5452 at x = 1). The derivative of 7^x is 7^x · ln 7 (ln 7 ≈ 1.9459; slope ≈ 13.6214 at x = 1). The derivative of 10^x is 10^x · ln 10 (ln 10 ≈ 2.3026; slope ≈ 23.0259 at x = 1). Notice the toll growing with the base — a bigger base climbs faster at the same height — and notice e’s special place in the row: when a = e, ln(a) = 1 and the toll vanishes, which is the entire reason e owns this corner of mathematics. The derivative of 5e^x, by contrast, needs no toll at all: 5e^x is a constant multiple of e^x, so it differentiates to 5e^x, slope 5e ≈ 13.5914 at x = 1.

The full table

Every function from the search tail, derived in one line. Values in the text above each passed an independent numeric check at the quoted point.

The 1/2x ambiguity — read the question before answering it

The derivative of 1/2x is the one entry in this table with two correct answers, because the typed string is genuinely ambiguous. Read as (1/2)·x — “half of x” — it is a line, and the derivative is 1/2. Read as 1/(2x) — “one over two-x” — it is a reciprocal, and the derivative is −1/(2x^2), by the chain rule on (2x)^-1. Written mathematics resolves this with the fraction bar’s length and position; typed search strings cannot. If your homework writes 1 over a long bar with 2x beneath it, you are in reciprocal country; if the 2 sits beneath the 1 alone, with x outside the fraction, it is the line. When in doubt, answer with the reading stated — “reading this as 1/(2x)…” — which examiners accept and search engines cannot.

The e^x neighbourhood: three chain-rule neighbours

Three near-relatives of the table’s exponential row are searched often enough to earn their derivations. The derivative of e^(2x) is 2e^(2x): the exponential reproduces itself, then the chain rule collects the inside derivative 2 — at x = 1 the slope is 2e^2 ≈ 14.7781, double the plain e^x height there, which is the toll made visible. The derivative of ln(2x) is the neighbourhood’s small surprise: the chain rule gives (1/(2x)) · 2, and the 2s cancel, leaving 1/x — identical to the derivative of ln(x) itself. Doubling the input shifts the log curve sideways without steepening it at all, and the algebra knows it before the graph does. And the sum of the two table rows behaves as the sum rule promises: the derivative of e^x + ln(x) is e^x + 1/x, slope e + 1 ≈ 3.7183 at x = 1. Neighbours like these are why the table is organised by family: meet the family once and its chain-rule cousins are recognisable on sight.

Spot-checks: two rows re-verified in the open

Trust in a reference table should be checkable from inside the table, so here are two rows re-derived by raw arithmetic, using the symmetric slope test — evaluate the original function a hair either side of a point and divide the rise by the run. For 1/x at x = 2: the values at 2.001 and 1.999 are 0.4997501249 and 0.5002501251, and (0.4997501249 − 0.5002501251) ÷ 0.002 = −0.2500000625 — the table’s −1/x^2, which at x = 2 is −0.25, to six decimal places of agreement. For x^3 at x = 2: the cubes at 2.001 and 1.999 are 8.012006001 and 7.988005999, and the quotient is 12.000001 against the table’s 3x^2 = 12. One part in a million of disagreement, shrinking as the step shrinks — the signature of a correct derivative, and the same test, run at any point of any row, is the reader’s own audit.

How these were verified

How to find derivative of a function — any function — comes down to two habits this table was built under. First, derive, do not recall: every row above is produced by a named rule from the function’s shape, so the method transfers to functions not in the table. Second, verify numerically: each derivative was evaluated at a probe point and compared against a symmetric difference quotient of the original function, and the two agreed in every row — the same slope test the hub page teaches for checking your own work. The derivative of a function you have never seen is not a memory problem; it is those two habits, applied once.