Derivative Rules
The core differentiation rules plus common derivatives — browse, filter, and copy.
Click any rule to copy it.
The core differentiation rules are the constant, power, constant-multiple, sum/difference, product, quotient, and chain rules. The power rule gives d/dx (xⁿ) = n·xⁿ⁻¹, and the chain rule handles compositions: d/dx f(g(x)) = f′(g(x))·g′(x). Combine these with the standard trig, exponential, and log derivatives to differentiate almost any function.
How the rules fit together
Differentiation is built from a small set of rules applied in layers. The power rule handles polynomials; the sum, constant-multiple, product, and quotient rules combine simpler pieces; and the chain rule wraps around any composition. Once you know the derivatives of the elementary functions — powers, sine and cosine, eˣ, and ln x — every other derivative follows by assembling these rules in the right order.
The elementary derivatives to memorise
A handful of standard results appear again and again: d/dx sin x = cos x, d/dx cos x = −sin x, d/dx eˣ = eˣ, and d/dx ln x = 1/x. The trigonometric derivatives come in matched pairs, and the co-functions (cos, cot, csc) all carry a leading minus sign. Keep the signs straight and the rest is bookkeeping.
Core differentiation rules
The seven structural rules that combine simpler derivatives.
| Rule | Formula |
|---|---|
| Constant | d/dx (c) = 0 |
| Power | d/dx (xⁿ) = n·xⁿ⁻¹ |
| Constant multiple | d/dx (c·f) = c·f′ |
| Sum / difference | (f ± g)′ = f′ ± g′ |
| Product | (f·g)′ = f′g + f g′ |
| Quotient | (f/g)′ = (f′g − f g′) / g² |
| Chain | d/dx f(g(x)) = f′(g(x))·g′(x) |
Common derivatives
Elementary functions — trigonometric, exponential, logarithmic, and inverse trig.
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
| cot x | −csc²x |
| sec x | sec x tan x |
| csc x | −csc x cot x |
| eˣ | eˣ |
| aˣ | aˣ ln a |
| ln x | 1/x |
| logₐ x | 1 / (x ln a) |
| arcsin x | 1 / √(1−x²) |
| arccos x | −1 / √(1−x²) |
| arctan x | 1 / (1+x²) |
The chain rule is the key to composites
Most real-world functions are compositions — one function inside another — and the chain rule is what unlocks them. To differentiate f(g(x)), take the derivative of the outer function evaluated at the inner one, then multiply by the derivative of the inner function: f′(g(x))·g′(x). For example, d/dx sin(x²) = cos(x²)·2x. Whenever you see a function tucked inside sine, an exponential, a root, or a power, reach for the chain rule and work from the outside in.