Derivative Table
Find common derivatives and differentiation rules in one searchable reference table. Browse power, exponential, logarithmic, trigonometric, and inverse functions, with formulas and conditions shown together.
Table of Common Derivatives
This derivative table lists standard functions beside their first derivatives. Use search or category filters to find a function quickly, then check its domain and any restrictions before applying the formula.
All formulas use real variables and principal inverse functions as specified. exp means the natural exponential, ln means natural logarithm, abs means absolute value, and ^ denotes a power. This is a symbolic reference, not a numerical derivative estimator or a free-form solver.
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| Name | Function f(x) | Derivative f′(x) | Conditions |
|---|---|---|---|
| ConstantConstants and Powers | c | 0 | c is constant; x ∈ ℝ. |
Details & verification: ConstantThe difference quotient is (c − c)/h = 0. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| IdentityConstants and Powers | x | 1 | x ∈ ℝ. |
Details & verification: IdentityThe difference quotient is h/h = 1. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| PowerConstants and Powers | x^n | n·x^(n-1) | Real n; x > 0. Integer powers extend wherever the function and derivative exist. n = 0 gives 0. |
Details & verification: PowerWrite xⁿ = exp(n ln x) for x > 0 and use the chain rule. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Square rootConstants and Powers | √(x) | 1/(2·√(x)) | x > 0; no finite derivative at x = 0. |
Details & verification: Square rootThe power rule with n = 1/2 gives (1/2)x⁻¹ᐟ². Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| ReciprocalConstants and Powers | 1/x | -1/x^2 | x ≠ 0. |
Details & verification: ReciprocalSet n = −1 in the power rule. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Natural exponentialExponential Functions | exp(x) | exp(x) | x ∈ ℝ. |
Details & verification: Natural exponentialThe limit (exp(h) − 1)/h = 1 leaves exp(x). Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| General exponentialExponential Functions | a^x | a^x·ln(a) | a > 0, a ≠ 1; x ∈ ℝ. |
Details & verification: General exponentialUse aˣ = exp(x ln a). Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Linear exponentialExponential Functions | exp(a·x) | a·exp(a·x) | a is a real constant; x ∈ ℝ. |
Details & verification: Linear exponentialThe inner derivative of ax supplies the factor a. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Natural logarithmLogarithmic Functions | ln(x) | 1/x | x > 0. |
Details & verification: Natural logarithmDifferentiate exp(ln x) = x to obtain exp(ln x)(ln x)′ = 1. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Logarithm of absolute valueLogarithmic Functions | ln(abs(x)) | 1/x | x ≠ 0; treat positive and negative intervals separately. |
Details & verification: Logarithm of absolute valueFor x < 0, d ln(−x)/dx = (−1)/(−x) = 1/x. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Base-a logarithmLogarithmic Functions | ln(x)/ln(a) | 1/(x·ln(a)) | x > 0; a > 0, a ≠ 1. |
Details & verification: Base-a logarithmChange of base gives ln x / ln a. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| SineTrigonometric Functions | sin(x) | cos(x) | x ∈ ℝ. Arguments are in radians. |
Details & verification: SineThe sine addition formula and the limits sin h/h → 1 and (cos h − 1)/h → 0 give cos x. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| CosineTrigonometric Functions | cos(x) | -sin(x) | x ∈ ℝ. Arguments are in radians. |
Details & verification: CosineApply the cosine addition formula; the remaining first-order term is −sin x. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| TangentTrigonometric Functions | tan(x) | 1/cos(x)^2 | cos x ≠ 0. Arguments are in radians. |
Details & verification: TangentThe quotient rule on sin x / cos x gives (cos²x + sin²x)/cos²x. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| CotangentTrigonometric Functions | cot(x) | -1/sin(x)^2 | sin x ≠ 0. Arguments are in radians. |
Details & verification: CotangentThe quotient rule on cos x / sin x gives −(sin²x + cos²x)/sin²x. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| SecantTrigonometric Functions | sec(x) | sec(x)·tan(x) | cos x ≠ 0. Arguments are in radians. |
Details & verification: SecantDifferentiate (cos x)⁻¹: sin x / cos²x = sec x tan x. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| CosecantTrigonometric Functions | csc(x) | -csc(x)·cot(x) | sin x ≠ 0. Arguments are in radians. |
Details & verification: CosecantDifferentiate (sin x)⁻¹: −cos x / sin²x = −csc x cot x. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| ArcsineInverse Trigonometric Functions | asin(x) | 1/√(1-x^2) | −1 < x < 1; principal range [−π/2, π/2]. Angles are in radians. |
Details & verification: ArcsineIf sin y = x, then cos y y′ = 1. On this branch cos y = √(1 − x²). Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| ArccosineInverse Trigonometric Functions | acos(x) | -1/√(1-x^2) | −1 < x < 1; principal range [0, π]. Angles are in radians. |
Details & verification: ArccosineIf cos y = x, then −sin y y′ = 1; sin y = √(1 − x²). Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| ArctangentInverse Trigonometric Functions | atan(x) | 1/(1+x^2) | x ∈ ℝ; principal range (−π/2, π/2). Angles are in radians. |
Details & verification: ArctangentDifferentiate tan y = x; sec²y = 1 + x². Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| ArccotangentInverse Trigonometric Functions | acot(x) | -1/(1+x^2) | x ∈ ℝ; arccot x ∈ (0, π). Angles are in radians. |
Details & verification: ArccotangentUse arccot x = π/2 − arctan x. This states the branch explicitly. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| ArcsecantInverse Trigonometric Functions | asec(x) | 1/(abs(x)·√(x^2-1)) | |x| > 1; arcsec x = arccos(1/x), range [0, π] excluding π/2. Angles are in radians. |
Details & verification: ArcsecantDifferentiate arccos(1/x); √(1 − 1/x²) = √(x² − 1)/|x|. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| ArccosecantInverse Trigonometric Functions | acsc(x) | -1/(abs(x)·√(x^2-1)) | |x| > 1; arccsc x = arcsin(1/x). Angles are in radians. |
Details & verification: ArccosecantDifferentiate arcsin(1/x); the absolute value comes from √(x²) = |x|. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Hyperbolic sineHyperbolic Functions | sinh(x) | cosh(x) | x ∈ ℝ. |
Details & verification: Hyperbolic sineDifferentiate (exp x − exp(−x))/2. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Hyperbolic cosineHyperbolic Functions | cosh(x) | sinh(x) | x ∈ ℝ. |
Details & verification: Hyperbolic cosineDifferentiate (exp x + exp(−x))/2; the second term changes sign. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Hyperbolic tangentHyperbolic Functions | tanh(x) | 1/cosh(x)^2 | x ∈ ℝ. |
Details & verification: Hyperbolic tangentThe quotient rule on sinh/cosh uses cosh² − sinh² = 1. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Hyperbolic cotangentHyperbolic Functions | coth(x) | -1/sinh(x)^2 | x ≠ 0. |
Details & verification: Hyperbolic cotangentThe quotient rule on cosh/sinh gives −1/sinh²x. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Hyperbolic secantHyperbolic Functions | sech(x) | -sech(x)·tanh(x) | x ∈ ℝ. |
Details & verification: Hyperbolic secantDifferentiate 1/cosh x to obtain −sinh x/cosh²x. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Hyperbolic cosecantHyperbolic Functions | csch(x) | -csch(x)·coth(x) | x ≠ 0. |
Details & verification: Hyperbolic cosecantDifferentiate 1/sinh x to obtain −cosh x/sinh²x. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Inverse hyperbolic sineInverse Hyperbolic Functions | asinh(x) | 1/√(1+x^2) | x ∈ ℝ. |
Details & verification: Inverse hyperbolic sineDifferentiate ln(x + √(1 + x²)); the positive square root cancels. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Inverse hyperbolic cosineInverse Hyperbolic Functions | acosh(x) | 1/√(x^2-1) | x > 1; principal acosh x ≥ 0. |
Details & verification: Inverse hyperbolic cosineFrom cosh y = x, sinh y = √(x² − 1) on the positive branch. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Inverse hyperbolic tangentInverse Hyperbolic Functions | atanh(x) | 1/(1-x^2) | |x| < 1. |
Details & verification: Inverse hyperbolic tangentDifferentiate (1/2)ln((1 + x)/(1 − x)). Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Inverse hyperbolic cotangentInverse Hyperbolic Functions | acoth(x) | 1/(1-x^2) | |x| > 1; real branch (1/2)ln|(x + 1)/(x − 1)|. |
Details & verification: Inverse hyperbolic cotangentIts logarithmic derivative is (1/2)(1/(x + 1) − 1/(x − 1)). Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Inverse hyperbolic secantInverse Hyperbolic Functions | asech(x) | -1/(x·√(1-x^2)) | 0 < x < 1; asech x = acosh(1/x). |
Details & verification: Inverse hyperbolic secantDifferentiate acosh(1/x) and use x > 0 to simplify the square root. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Inverse hyperbolic cosecantInverse Hyperbolic Functions | acsch(x) | -1/(abs(x)·√(1+x^2)) | x ≠ 0; acsch x = asinh(1/x). |
Details & verification: Inverse hyperbolic cosecantDifferentiate asinh(1/x); √(1 + 1/x²) = √(1 + x²)/|x|. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. The reverse relationship is an antiderivative: see integration formulas and conditions. | |||
| Sum and differenceDifferentiation Rules | f(x) ± g(x) | f′(x) ± g′(x) | f and g differentiable. |
Details & verification: Sum and differenceThe difference quotient distributes over addition and subtraction. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. | |||
| Constant multipleDifferentiation Rules | c f(x) | c f′(x) | c constant; f differentiable. |
Details & verification: Constant multipleFactor c out of the difference quotient. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. | |||
| ProductDifferentiation Rules | f(x)g(x) | f′(x)g(x) + f(x)g′(x) | f and g differentiable. |
Details & verification: ProductAdd and subtract f(x+h)g(x) in the numerator of the difference quotient. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. | |||
| QuotientDifferentiation Rules | f(x)/g(x) | (f′(x)g(x) − f(x)g′(x))/g(x)² | f and g differentiable; g(x) ≠ 0. |
Details & verification: QuotientApply the product rule to f times g⁻¹ and differentiate the reciprocal. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. | |||
| ChainDifferentiation Rules | f(g(x)) | f′(g(x))g′(x) | g differentiable at x; f differentiable at g(x). |
Details & verification: ChainThe first-order changes multiply: Δf ≈ f′(g(x))Δg and Δg ≈ g′(x)Δx. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. | |||
| Reciprocal ruleDifferentiation Rules | 1/f(x) | −f′(x)/f(x)² | f differentiable; f(x) ≠ 0. |
Details & verification: Reciprocal ruleDifferentiate f(x) times its reciprocal, which equals 1. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. | |||
| Implicit differentiationDifferentiation Rules | F(x, y) = 0 | dy/dx = −Fₓ/Fᵧ | F continuously differentiable near the point; Fᵧ ≠ 0. |
Details & verification: Implicit differentiationDifferentiate along the curve: Fₓ + Fᵧ y′ = 0, then solve for y′. This is a local rule, not an arbitrary solver. Reference formula or deriving rule. Use the stated domain and branch, not just the algebraic expression. | |||
Basic Derivative Rules
Differentiation rules let you combine simpler results to differentiate more complicated functions.
The power rule states that the derivative of xⁿ is nxⁿ⁻¹ wherever the function and rule apply.
The product rule differentiates a product of two functions, while the quotient rule handles a ratio with a nonzero denominator.
The chain rule is essential for composite functions: if y = f(g(x)), then y′ = f′(g(x))g′(x).
Table of Trigonometric Derivatives
The derivatives of sine, cosine, tangent, and the other trigonometric functions depend on their arguments being measured in radians for the standard formulas.
For example:
d/dx[sin x] = cos x
d/dx[cos x] = −sin x
d/dx[tan x] = sec² x
For functions such as sin(g(x)), apply the chain rule as well.
Exponential and Logarithmic Derivatives
The derivative of eˣ is eˣ. More generally, the derivative of aˣ is aˣ ln(a) for a positive base other than 1.
The derivative of ln|x| is 1/x for x ≠ 0. The logarithm base and function domain must be considered when applying logarithmic derivative formulas.
How to Use a Derivative Table
First, identify the function family. Next, find the matching formula. If your function contains a sum, product, quotient, or composition, combine the table entry with the corresponding differentiation rule.
Always check that the expression is defined at the point where the derivative is needed.
Step-by-Step Derivative Examples
Example 1 — Polynomial
f(x) = 3x² + 2x − 5
Differentiate term by term:
f′(x) = 6x + 2.
Example 2 — Chain rule
f(x) = sin(x²)
The outer function is sine and the inner function is x²:
f′(x) = cos(x²) × 2x = 2x cos(x²).
Example 3 — Logarithm
f(x) = ln|x|
f′(x) = 1/x, for x ≠ 0.
Frequently Asked Questions
What is a derivative table?
A derivative table is a reference listing common functions and their derivatives. It helps you recognize standard results and apply differentiation rules without deriving each formula from the limit definition.
How do I use a table of derivatives?
Find your function in the table and read its derivative. If the function is a combination or composition of simpler expressions, use the appropriate sum, product, quotient, or chain rule.
What is the derivative of x to the power n?
The power rule gives d/dx[xⁿ] = nxⁿ⁻¹ wherever the real-valued expression and derivative are defined.
What is the derivative of sin(x)?
The derivative of sin(x) is cos(x) when x is measured in radians.
What is the derivative of ln(x)?
For x > 0, the derivative of ln(x) is 1/x. The derivative of ln|x| is also 1/x wherever x ≠ 0.
What is the chain rule in differentiation?
The chain rule gives the derivative of a composite function. If y = f(g(x)), then y′ = f′(g(x))g′(x), provided the relevant derivatives exist.
Are derivative formulas always valid for every x?
No. Formulas may require domain restrictions, nonzero denominators, positive logarithm arguments, or other conditions. The derivative itself may also fail to exist at a discontinuity or corner.
Is finding a derivative from a table of values the same task?
No. A derivative formula table lists symbolic derivatives. A numerical table of x and y values is usually used to estimate a derivative from changes in the data, often with finite differences.
Sources and verification
Lamar University — Common Derivatives and Integrals supplies standard formulas and rules. Each entry states its mathematical justification and branch conventions; derivative identities are independently checked during development, with negative intervals checked separately where absolute values matter.
Integral Table reverses the derivative relationship. For derivative operations in transforms, see the Laplace Transform Table. Browse all tables or read our methodology.