Formula · Calculus
Derivative Rules: The Complete Table
Two short tables cover almost everything a first calculus course asks for. The second one is worth memorising; the first is worth understanding.
The structural rules
These say how to differentiate a combination once you can differentiate its parts. They are the derivative rules that do the real work.
| Name | Rule |
|---|---|
| Constant | d/dx (c) = 0 |
| Constant multiple | d/dx (c·u) = c·u′ |
| Sum and difference | d/dx (u ± v) = u′ ± v′ |
| Power rule | d/dx (xⁿ) = n·xⁿ⁻¹ |
| Product rule | d/dx (uv) = u′v + uv′ |
| Quotient rule | d/dx (u/v) = (u′v − uv′) / v² |
| Chain rule | d/dx f(g(x)) = f′(g(x)) · g′(x) |
The power rule holds for every real exponent, including negative and fractional ones. That single fact removes a lot of memorisation: √x is x^½ and 1/x³ is x⁻³, and both then need nothing new.
The standard functions
Each of these has a chain form, which is the same rule with an inside function and its derivative attached.
| f(x) | f′(x) | With an inside function u |
|---|---|---|
| sin x | cos x | cos u · u′ |
| cos x | −sin x | −sin u · u′ |
| tan x | sec² x | sec² u · u′ |
| eˣ | eˣ | e^u · u′ |
| aˣ | aˣ ln a | a^u ln a · u′ |
| ln x | 1/x | u′/u |
| √x | 1/(2√x) | u′/(2√u) |
The minus sign on cos is the one most often lost, and it flips the whole answer. A quick memory hook: the derivatives of the 'co' functions — cos, cot, csc — all carry a minus.
Inverse trig and hyperbolic
Less often needed, and rarely worth memorising when a table is available.
| f(x) | f′(x) |
|---|---|
| arcsin x | 1/√(1 − x²) |
| arccos x | −1/√(1 − x²) |
| arctan x | 1/(1 + x²) |
| sinh x | cosh x |
| cosh x | sinh x |
| tanh x | sech² x |
Notice that arcsin and arccos differ only by a sign, and that arctan has no square root at all — which is why it turns up so often as the answer to an integral.
What is not on these tables
Two gaps are worth naming, because they are where a first course stops.
- Implicit differentiation. When y is not written explicitly in terms of x — a circle, x² + y² = 25 — you differentiate both sides and treat y as a function of x, which brings a dy/dx along by the chain rule.
- Partial derivatives. When a quantity depends on several variables, you differentiate with respect to one and hold the others fixed. The rules above are unchanged; only the bookkeeping is new.
Neither needs a new table. Both are the same rules applied with one extra piece of care about what is a function of what.
Why eˣ is its own derivative
This is not a coincidence to be memorised but the definition of e. Among all exponential functions aˣ, the derivative is aˣ ln a — so the growth rate is proportional to the current value, with ln a as the constant of proportionality.
e is the base where that constant is exactly 1. The function then grows at precisely the rate of its own size, which is why e appears in every model of natural growth and decay.
The chain form is the general form
Every entry in the tables above is the special case where the inside function is a bare x. The chain form is what you actually use most of the time.
| Simple form | Chain form | Example |
|---|---|---|
| (sin x)′ = cos x | (sin u)′ = cos u · u′ | sin(3x) → 3cos(3x) |
| (eˣ)′ = eˣ | (e^u)′ = e^u · u′ | e^(x²) → 2x e^(x²) |
| (ln x)′ = 1/x | (ln u)′ = u′/u | ln(x² + 1) → 2x/(x² + 1) |
Learning the chain form rather than the simple one saves relearning each rule later. The simple version falls out of it whenever u is x, because then u′ is 1.
Notation, and the second derivative
Two notations mean the same thing, and both appear constantly.
| Lagrange | Leibniz | Reads as |
|---|---|---|
| f′(x) | dy/dx | the first derivative |
| f″(x) | d²y/dx² | the second derivative |
| f⁽ⁿ⁾(x) | dⁿy/dxⁿ | the nth derivative |
Leibniz notation names both variables, which matters whenever it could be ambiguous what is changing with respect to what. The second derivative is simply the derivative of the derivative, and it describes how the curve bends.
Questions about derivative rules
What are the main derivative rules?
The power rule, the constant multiple and sum rules, and the three structural rules for combinations: product, quotient and chain.
Does the power rule work for negative exponents?
Yes, and for fractional ones. x⁻² gives −2x⁻³ and x^½ gives ½x^−½, which is 1/(2√x).
Which derivatives have a minus sign?
The 'co' functions: cos gives −sin, cot gives −csc², and csc gives −csc·cot. Arccos also carries one.
Why is the derivative of eˣ equal to eˣ?
Because e is defined as the base whose growth rate equals its current value. For any other base a, the derivative is aˣ ln a, and ln e is 1.