liminfoDerivatives Reference Table
Free web tool: Derivatives Reference Table
About Derivatives Reference Table
The Derivatives Reference Table is a searchable, filterable cheat sheet containing 33 differentiation rules organized across five categories: Basic, Trigonometric, Inverse Trig, Exponential/Log, and Hyperbolic. Each entry displays the function f(x) alongside its derivative d/dx in a clean monospace format. You can search by function name or derivative expression, and filter by category to instantly narrow down the formula you need.
Calculus students and engineers frequently need to look up derivative rules during problem-solving, exam preparation, or implementation of numerical algorithms. This reference covers the complete set of elementary function derivatives that appear in standard single-variable calculus courses: the 8 basic differentiation rules (constant, power, scalar multiple, sum, product, quotient, and chain rules), all 6 trigonometric derivatives, all 6 inverse trigonometric derivatives, 5 exponential and logarithmic derivatives including general base forms, and 9 hyperbolic and inverse hyperbolic function derivatives.
The tool is built entirely client-side: the full set of 33 formulas is embedded in the component, and real-time filtering uses JavaScript array methods with no server requests. This means the reference loads instantly, works offline, and can be bookmarked for quick access during study sessions or while working on calculus problems.
Key Features
- Contains 33 differentiation rules covering all elementary function families
- Five filter categories: Basic, Trigonometric, Inverse Trig, Exponential/Log, Hyperbolic
- Live search by function name or derivative expression
- Formula count display updates dynamically as you filter
- Monospace font rendering for clear mathematical notation
- Category badge on each formula card for quick visual identification
- Hover highlight on each formula row for easy reading
- 100% client-side — loads instantly, works offline, no sign-up required
Frequently Asked Questions
What differentiation rules are included in this reference?
The reference includes 33 rules across five categories: Basic (constant rule, identity, power rule, scalar multiple, sum rule, product rule, quotient rule, chain rule), Trigonometric (sin, cos, tan, cot, sec, csc), Inverse Trig (arcsin, arccos, arctan, arccot, arcsec, arccsc), Exponential/Log (eˣ, aˣ, ln x, log_a x, xˣ), and Hyperbolic (sinh, cosh, tanh, coth, sech, csch, arcsinh, arccosh, arctanh).
What is the chain rule and when do I use it?
The chain rule states that d/dx[f(g(x))] = f'(g(x)) · g'(x). Use it whenever you differentiate a composite function — one function nested inside another. For example, d/dx[sin(x²)] requires the chain rule: the outer function is sin(u) with derivative cos(u), and the inner function is x² with derivative 2x, giving cos(x²) · 2x.
What is the product rule?
The product rule states that d/dx[f(x)·g(x)] = f'(x)g(x) + f(x)g'(x). Use it when differentiating the product of two functions that both depend on x. A common mnemonic is "first times derivative of second, plus second times derivative of first."
What is the quotient rule?
The quotient rule states that d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]². Use it when differentiating a fraction where both numerator and denominator are functions of x. Note the minus sign in the numerator (unlike the product rule which uses addition).
Why is the derivative of eˣ equal to itself?
The exponential function eˣ is the unique function (up to scalar multiples) that equals its own derivative. This property follows from the definition of e as the base for which the slope of the tangent to the curve at any point equals the y-value at that point. It is the foundation of exponential growth and decay models in science and engineering.
What is the derivative of ln(x)?
The derivative of ln(x) is 1/x, valid for x > 0. More generally, by the chain rule, d/dx[ln(u)] = u'/u. This is one of the most frequently used derivatives in integration (integration by parts, partial fractions) and in logarithmic differentiation techniques.
How do hyperbolic function derivatives differ from trigonometric derivatives?
Hyperbolic functions (sinh, cosh, tanh, etc.) are defined using exponentials: sinh(x) = (eˣ - e⁻ˣ)/2 and cosh(x) = (eˣ + e⁻ˣ)/2. Their derivatives closely mirror trig derivatives but without alternating signs: d/dx[sinh(x)] = cosh(x) and d/dx[cosh(x)] = sinh(x). Note that tanh'(x) = sech²(x), parallel to tan'(x) = sec²(x).
Is this derivatives reference table free?
Yes, completely free. The entire formula set is embedded in the page and processed locally in your browser. No account, no download, and no server connection is needed. You can bookmark the page for quick offline access during your study sessions.