Polynomial
Differentiate f(x) = 3x4 − 2x + 7.
f′(x) = 12x3 − 2. The derivative of the constant is zero.
Derivative Rules
A one-page reference for differentiating common functions. Rules are listed from most used to least used, followed by worked examples and the mistakes students make most often.
Core rules
| Function | Derivative |
|---|---|
c (constant) | 0 |
x | 1 |
xn | n xn−1 |
ex | ex |
ax | ax ln a |
ln x | 1/x |
loga x | 1 / (x ln a) |
sin x | cos x |
cos x | −sin x |
tan x | sec2 x |
cot x | −csc2 x |
sec x | sec x tan x |
csc x | −csc x cot x |
arcsin x | 1 / √(1 − x2) |
arccos x | −1 / √(1 − x2) |
arctan x | 1 / (1 + x2) |
Combination rules
| Rule | Formula |
|---|---|
| Sum | (f + g)′ = f′ + g′ |
| Difference | (f − g)′ = f′ − g′ |
| Constant multiple | (c f)′ = c f′ |
| Product | (f g)′ = f′g + fg′ |
| Quotient | (f/g)′ = (f′g − fg′) / g2 |
| Chain | (f(g(x)))′ = f′(g(x)) · g′(x) |
Using the chain rule
Identify the outer function and the inner function. Differentiate the outside, keep the inside unchanged, then multiply by the derivative of the inside.
d/dx [sin(3x)] = cos(3x) · 3 = 3 cos(3x)
Common mistakes
- Forgetting the chain rule when the inside is not just
x. - Applying the power rule to
exorax. - Writing the derivative of
ln xasln x / xor1 / ln x. - Dropping the negative sign for
cos xandcot x. - Differentiating a product term by term instead of using the product rule.
Worked examples
Chain rule
Differentiate f(x) = sin(2x).
Outer function: sin. Inner function: 2x.
f′(x) = cos(2x) · 2 = 2 cos(2x).
Product rule
Differentiate f(x) = x2 ex.
f′(x) = 2x ex + x2 ex = x ex(2 + x).
FAQ
Is the derivative of ln x the same as ln|x|?
Yes. For x ≠ 0, d/dx ln|x| = 1/x.
Why is the derivative of a constant zero?
A constant does not change, so its rate of change is zero.
Do I need to memorize every rule?
Memorize the core table and the combination rules. The rest can be derived quickly.
Last updated November 1, 2026. Formulas are checked against standard references. Read the editorial policy.