The rules
| Rule | Formula |
|---|---|
| Constant | d/dx c = 0 |
| Constant multiple | (c·f)′ = c·f′ |
| Sum and difference | (f ± g)′ = f′ ± g′ |
| Power rule | d/dx xⁿ = n·xⁿ⁻¹ (any real n) |
| Product rule | (fg)′ = f′g + fg′ |
| Quotient rule | (f/g)′ = (f′g − fg′)/g² |
| Chain rule | d/dx f(g(x)) = f′(g(x)) · g′(x) |
Table of common derivatives
| Function f(x) | Derivative f′(x) |
|---|---|
| c (a constant) | 0 |
| x | 1 |
| xⁿ | n·xⁿ⁻¹ |
| 1/x | −1/x² |
| √x | 1/(2√x) |
| eˣ | eˣ |
| aˣ | aˣ · ln a |
| ln x | 1/x (x > 0) |
| ln |x| | 1/x (x ≠ 0) |
| log_b x | 1/(x · ln b) |
| 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) |
| sinh x | cosh x |
| cosh x | sinh x |
Every entry can be combined with the chain rule: replace x with an inner function u and multiply by u′. For example, d/dx e^(x²) = e^(x²) · 2x.
The derivative of tan x
Write tan x = sin x / cos x and use the quotient rule:
d/dx tan x = (cos x · cos x − sin x · (−sin x)) / cos² x = (cos² x + sin² x) / cos² x = 1/cos² x = sec² x.
The numerator simplifies because cos² x + sin² x = 1. With the chain rule, d/dx tan(2x) = 2 sec²(2x). At x = π/4, the slope of tan x is sec²(π/4) = 2.
The derivative of ln x
d/dx ln x = 1/x, for x > 0. One way to see it: if y = ln x, then eʸ = x. Differentiate both sides with respect to x: eʸ · y′ = 1, so y′ = 1/eʸ = 1/x.
- d/dx ln(3x) = 3/(3x) = 1/x. It's the same as ln x because ln(3x) = ln 3 + ln x, and ln 3 is a constant.
- d/dx ln(x² + 1) = 2x/(x² + 1), by the chain rule.
- d/dx ln|x| = 1/x for every x ≠ 0, which is why ∫(1/x) dx = ln|x| + C.
- d/dx (x² ln x) = 2x ln x + x²·(1/x) = x(2 ln x + 1), by the product rule.
For other bases, log_b x = ln x / ln b, so d/dx log_b x = 1/(x ln b). See the change-of-base formula.
The derivative of 1/x
Rewrite 1/x as x⁻¹ and use the power rule: d/dx x⁻¹ = −1 · x⁻² = −1/x². The same idea gives d/dx (1/x²) = d/dx x⁻² = −2x⁻³ = −2/x³, and d/dx √x = d/dx x^(1/2) = ½x^(−1/2) = 1/(2√x). Rewriting roots and fractions as powers is covered in exponent rules.
Worked example with several rules
Differentiate f(x) = x² sin x.
- It's a product, so use the product rule with u = x² and v = sin x.
- u′ = 2x (power rule) and v′ = cos x (table).
- f′(x) = 2x sin x + x² cos x.
Where the trig derivatives come from
The derivatives of sin x and cos x come from the limit definition and the limit sin h / h → 1 as h → 0 (proved with the squeeze theorem). The rest follow from the quotient rule: cot x = cos x / sin x, sec x = 1/cos x, csc x = 1/sin x. They're only this simple when x is in radians; see degrees and radians. For the inverse trig rows, see derivatives of inverse trig functions.
Differentiate anything step by step with the derivative calculator, and reverse the process with the integral calculator.
Sources
- OpenStax Calculus Volume 1, 3.3: Differentiation rules
- OpenStax Calculus Volume 1, 3.5: Derivatives of trigonometric functions
- OpenStax Calculus Volume 1, 3.9: Derivatives of exponential and logarithmic functions
Every worked example on this page was checked independently with SymPy.