Derivative Rules: Complete Cheat Sheet with Examples
Quick answer: the core derivative rules are the power rule (xⁿ)′ = nxⁿ⁻¹, the product rule (uv)′ = u′v + uv′, the quotient rule (u/v)′ = (u′v − uv′)/v² and the chain rule f(g(x))′ = f′(g(x))·g′(x).
Derivative rules table
| Rule | Formula | Example |
|---|---|---|
| Constant | (c)′ = 0 | (7)′ = 0 |
| Power | (xⁿ)′ = n·xⁿ⁻¹ | (x⁵)′ = 5x⁴ |
| Constant multiple | (c·f)′ = c·f′ | (3x²)′ = 6x |
| Sum | (f + g)′ = f′ + g′ | (x² + x)′ = 2x + 1 |
| Product | (uv)′ = u′v + uv′ | (x sin x)′ = sin x + x cos x |
| Quotient | (u/v)′ = (u′v − uv′)/v² | (1/x)′ = −1/x² |
| Chain | f(g(x))′ = f′(g(x))·g′(x) | (sin 3x)′ = 3 cos 3x |
| Exponential | (eˣ)′ = eˣ | (e^(2x))′ = 2e^(2x) |
| Logarithm | (ln x)′ = 1/x | (ln 5x)′ = 1/x |
| Sine / cosine | (sin x)′ = cos x, (cos x)′ = −sin x | (cos x²)′ = −2x sin x² |
| Tangent | (tan x)′ = sec² x | (tan 2x)′ = 2 sec² 2x |
The power rule
Bring the exponent down and subtract one. It also works for negative and fractional powers: √x = x^(1/2), so (√x)′ = ½x^(−1/2) = 1/(2√x).
The product rule
Use it when two functions of x are multiplied. For f(x) = x³ sin x: u = x³, v = sin x, so f′(x) = 3x² sin x + x³ cos x.
The quotient rule
A handy memory aid: “low d-high minus high d-low, over the square of what’s below”. For f(x) = x / (x + 1): f′(x) = [(1)(x + 1) − x(1)] / (x + 1)² = 1 / (x + 1)².
The chain rule
Use the chain rule whenever there is a function inside another function. Differentiate the outside, keep the inside, then multiply by the derivative of the inside.
Example: f(x) = (2x + 1)⁵ → f′(x) = 5(2x + 1)⁴ · 2 = 10(2x + 1)⁴.
Common mistakes
- Forgetting the chain rule factor, e.g. writing (sin 3x)′ = cos 3x instead of 3 cos 3x.
- Treating the product rule as (uv)′ = u′v′.
- Mixing up the order in the quotient rule numerator.
Practise with any function in our free derivative calculator, which lists the rules used for every step.