Derivative Calculator
Fast, accurate, and free online Derivative Calculator tool that runs directly in your browser.
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The calculator calculates the derivative polynomial (the sum of the terms a·xⁿ) using the power rule. For functions such as sin, ln and eˣ, use the derivative table below.
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Other tools you may find usefulDerivative of a function - a basic concept of mathematical analysis
The derivative of a function is one of the most fundamental concepts in mathematics, physics and engineering sciences. It determines the rate of change of the function's value in relation to the change of its arguments. Geometrically, the derivative of a function at a given point is equal to the slope of the tangent to the graph of that function at that point - it tells us how steeply the graph "rises" or "falls". In turn, in physics, the derivative has a direct interpretation as instantaneous velocity (which is the derivative of distance with respect to time) or acceleration (derivative of speed with respect to time). The operation of calculating the derivative is called differentiation.
Our website offers a complete list of formulas for derivatives of elementary functions and the most important rules of differentiation (derivatives of sum, difference, product and quotient), making it easier to learn higher mathematics.
Basic rules for differentiating functions
Differentiation of complex functions is based on several iron-clad mathematical rules:
- Derivative of the sum and difference:
(f(x) ± g(x))' = f'(x) ± g'(x) - Derivative of the product:
(f(x) × g(x))' = f'(x) × g(x) + f(x) × g'(x) - Derivative of the quotient:
(f(x) / g(x))' = (f'(x) × g(x) - f(x) × g'(x)) / (g(x))² - Derivative of a composite function:the derivative of the outer function multiplied by the derivative of the inner function.
Table of formulas: derivatives of elementary functions
The table below presents a list of formulas for the derivatives of the most common mathematical functions:
| Name of the function | Formula of the function f(x) | Formula of the derivative of f'(x) | Domain and notes |
|---|---|---|---|
| Constant function | c (constant) | 0 | The derivative of a constant number is always zero. |
| Linear function (x) | x | 1 | The slope is 1. |
| Power function | xn | n × xn-1 | The basic formula used in polynomials. |
| Exponential function (e) | ex | ex | The function exis its own derivative. |
| Exponential function (a) | ax | ax× ln(a) | For a > 0. |
| Natural logarithm | ln(x) | 1 / x | For x > 0. |
| Sine function | sin(x) | cos(x) | The angle given in radians. |
| Cosine function | cos(x) | -sin(x) | The angle given in radians. |
Frequently asked questions (FAQ)
What is the derivative of a function?
This is a measure of the rate at which a function's value changes as its arguments change. Geometrically, it corresponds to the slope of the tangent to the graph of the function at a given point.
What is the formula for the derivative of a power function?
The formula is (xn)' = n × xn-1(e.g. the derivative of x³ is 3x²).
How to calculate the derivative of the product of two functions?
The formula applies: (u × v)' = u' × v + u × v'. This means that we multiply the derivative of the first function by the second function unchanged, and then add the first function unchanged multiplied by the derivative of the second.
What is the second derivative and what is it used for?
The second derivative is the derivative of the derivative of a given function. In mathematics, it is used to study the concavity and convexity of the graph of a function and to find the points of inflection of the graph.
Does every function have a derivative?
No. For a function to have a derivative at a given point, it must be continuous at that point and cannot have "sharp bends" (the graph must be smooth). An example of a continuous function that has no derivative at x = 0 is the absolute value function f(x) = |x|.