In calculus, a derivative measures a function’s instantaneous rate of change at a particular input. It is defined as the limit of average rates of change over shrinking intervals; when that limit exists, it is also the slope of the graph’s tangent line at that point.
How the derivative is defined
For a function f defined around a, the derivative at a is
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f′(a) = limh→0 [f(a + h) − f(a)] / h, provided the limit exists.
The fraction is the average rate of change between the inputs a and a + h. Geometrically, it is the slope of a secant line through the graph points (a, f(a)) and (a + h, f(a + h)). As h approaches zero, the second point moves toward the first. If the secant slopes approach one finite value, that value is the derivative and the tangent-line slope. This limit definition—not merely using a very small interval—is what makes the rate instantaneous. OpenStax, Calculus Volume 1, section 3.1
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What a derivative tells you
At a point: the direction and rate of change
A positive derivative means the function is increasing locally at that input; a negative derivative means it is decreasing locally. A derivative of zero means the graph has a horizontal tangent there, assuming the derivative exists. The derivative’s units are output units divided by input units, such as meters per second when position in meters is measured against time in seconds.
Across inputs: the derivative function
The notation f′ can refer to a new function that gives the derivative at every input where the defining limit exists. Saying that f is differentiable at a means its derivative exists at that input. OpenStax, Calculus Volume 1, section 3.2
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In motion: position becomes velocity
If s(t) describes position over time, then s′(t) is instantaneous velocity. Average velocity over a time interval is the change in position divided by the elapsed time; instantaneous velocity is the limiting rate at one moment. Derivatives also help describe acceleration, marginal profit functions, and biological growth rates. OpenStax, Calculus Volume 1, section 3.1 OpenStax, Calculus Volume 1, Chapter 3 Key Concepts
How to read derivative notation
f′(a) is read “f-prime of a” or “the derivative of f evaluated at a.” The notation dy/dx is another common way to represent a derivative. In elementary calculus, it denotes a rate of change defined through a limit; it is not ordinary division of two independent finite changes.
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When a derivative does not exist
The limit in the definition must approach one finite value. It may fail to do so at a sharp corner, cusp, discontinuity, or vertical tangent, so a function need not have a derivative at every point. Differentiability implies continuity, but continuity does not guarantee differentiability: a graph can be continuous and still have a corner. OpenStax, Calculus Volume 1, section 3.2
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Calculus derivatives and financial derivatives are different
“Derivative” also has a meaning in finance, where it refers to a financial instrument rather than a function’s slope or rate of change. The calculus definition above does not define financial products; readers asking about those should look for a finance-specific explanation.
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