The derivative at a point f′(a) is a number—one slope at one location. But the definition works at every point where the limit exists, and letting the input vary produces a new function f′(x) that maps each x to the slope of f at that x.
This shift in perspective is fundamental. The derivative function f′ has its own domain, its own graph, its own behavior. It can be continuous or discontinuous, positive or negative, increasing or decreasing. It can itself be differentiated, producing higher-order derivatives.
Understanding f′ as a function—not just a computation applied at isolated points—connects the local information encoded in slopes to global information about the shape of f.
The Derivative Function Defined
Fix a function f and consider the limit
f′(x)=h→0limhf(x+h)−f(x)
evaluated not at a single point a but at a variable x. Wherever this limit produces a finite value, f′(x) is defined. The result is a function f′ whose input is x and whose output is the slope of f at x.
The domain of f′ is a subset of the domain of f. Every point where f is differentiable belongs to the domain of f′; every point where the limit fails—corners, cusps, vertical tangents, discontinuities—is excluded. For a polynomial, the domains of f and f′ coincide. For f(x)=∣x∣, the domain of f is all reals, but x=0 is missing from the domain of f′.
Computing the Derivative Function from the Definition
Applying the limit definition with x as a free variable produces an expression in x—the derivative function in closed form.
The result f′(x)=2x is a function: input x, output the slope of the parabola at that x. At x=3, the slope is 6. At x=−1, the slope is −2.
Once differentiation rules are established, this limit computation becomes unnecessary for standard functions. But the limit definition remains the foundation—every rule is proven from it, and nonstandard functions may require returning to it directly.
Graphing f' from f
The graph of f′ can be read directly from the graph of f without computing any formula.
Where f is increasing, f′ is positive—the graph of f′ lies above the x-axis. Where f is decreasing, f′ is negative—the graph of f′ lies below the x-axis. Where f has a local extremum, f′ crosses zero (or is undefined).
Steep sections of f correspond to large values of ∣f′∣. Flat sections of f correspond to f′ near zero. A straight segment of f with constant slope corresponds to f′ being constant—a horizontal segment on the graph of f′.
Where f is concave up, the slope is increasing, so f′ is an increasing function on that interval. Where f is concave down, f′ is decreasing. Inflection points of f—where concavity changes—correspond to local extrema of f′.
Graphing f from f'
The reverse problem—reconstructing f from f′—is possible up to a vertical shift.
Where f′ is positive, f is rising. Where f′ is negative, f is falling. Where f′ crosses zero from positive to negative, f has a local maximum. Where f′ crosses zero from negative to positive, f has a local minimum.
The magnitude of f′ controls steepness: large positive values of f′ mean f is climbing steeply; values of f′ near zero mean f is nearly flat. Where f′ is increasing, f is concave up. Where f′ is decreasing, f is concave down.
The one piece of information f′ does not determine is the vertical position. The derivative of f(x)=x2 and the derivative of g(x)=x2+5 are identical: both equal 2x. The graph of f can be shifted vertically by any constant without changing f′. This ambiguity is resolved only when a specific function value is known—an initial condition.
The Relationship Between f and f'
The derivative function f′ encodes the shape of f completely, up to vertical translation. The correspondence runs in both directions:
Zeros of f′ correspond to horizontal tangents on f—local extrema or plateaus. Sign changes of f′ correspond to direction changes of f. Extrema of f′ correspond to inflection points of f, where the bending direction reverses.
If f′ is itself a smooth, well-behaved function, then f is smooth as well. If f′ has jumps or singularities, those indicate abrupt changes in the slope of f—corners, cusps, or worse.
This relationship extends to higher-order derivatives. The function f′′ relates to f′ in the same way that f′ relates to f: the second derivative is the derivative of the derivative, and the correspondence between zeros, sign changes, and extrema repeats at each level.
The Derivative as Rate of Change
When f models a quantity that varies with input x, the derivative function f′(x) models how fast that quantity changes at each value of x.
In kinematics, if s(t) is position as a function of time, then s′(t) is velocity—a function that gives the speed and direction of motion at every instant. The velocity function can be graphed, analyzed for zeros (stops), sign changes (reversals), and extrema (maximum speed). Differentiating again yields s′′(t), acceleration, which is itself a function of time.
In economics, if C(x) is the cost of producing x units, then C′(x) is the marginal cost—the rate at which cost changes per additional unit. The marginal cost function reveals whether production costs are accelerating or stabilizing.
In each context, the derivative function provides a running commentary on how the original quantity evolves—not at a single frozen moment, but across the entire domain.
Continuity of the Derivative
A natural question: if f is differentiable everywhere, must f′ be continuous?
The answer is no. Consider f(x)=x2sin(1/x) for x=0 and f(0)=0. This function is differentiable at every real number, including x=0, yet f′ oscillates near the origin and is discontinuous there. Differentiability of f does not guarantee continuity of f′.
However, derivatives satisfy a strong structural constraint. Darboux's theorem states that f′ always has the intermediate value property: if f′ takes values A and B on an interval, it takes every value between A and B somewhere on that interval. This means f′ cannot have jump discontinuities. The only discontinuities possible for a derivative are oscillatory ones—the kind seen in the example above.
Functions whose derivatives are continuous everywhere form the class C1. Most functions encountered in standard calculus belong to C1 or higher smoothness classes, but the distinction matters in theoretical work and in the study of differentiability.