The derivative of a function measures its instantaneous rate of change — how quickly the output changes as the input changes, at a single exact point. Geometrically, it's the slope of the tangent line to the function's graph. The derivative is defined as a limit and forms one half of calculus (the other being the integral).
When you're driving and glance at your speedometer, it shows your speed at that exact instant — not your average speed over the whole trip. But speed is normally distance divided by time — how do you calculate that for a single, zero-length instant? This is exactly the problem the derivative solves.
Picture a curved graph. At any single point, you can draw a straight line that just touches the curve there — the tangent line. The derivative at that point is the slope of that tangent line. Where the curve is steep, the derivative is large; where the curve is flat, the derivative is near zero; where the curve is going downward, the derivative is negative.
This is the limit of the "average rate of change" over an interval of length h, as h shrinks to zero. If this limit exists, f is differentiable at x.
| Function | Derivative |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| sin(x) | cos(x) |
| cos(x) | −sin(x) |
| eˣ | eˣ |
| ln(x) | 1/x |
The derivative of a derivative is the second derivative, written f″(x) or d²y/dx². If f represents position, f′ is velocity and f″ is acceleration. Higher derivatives continue similarly (jerk, snap, etc., in physics).
If a function is differentiable at a point, it must be continuous there — but the converse is false. The classic counterexample is f(x) = |x| at x = 0: continuous, but not differentiable (the tangent line direction changes abruptly, giving no well-defined slope).
Isaac Newton developed calculus (his "method of fluxions") around 1665–1666 but did not publish for decades. Gottfried Wilhelm Leibniz developed calculus independently around 1674–1676, publishing first (1684). This led to a bitter priority dispute between Newton's supporters and Leibniz's, played out over decades and involving accusations of plagiarism on both sides. Modern historical consensus (following detailed manuscript analysis) is that both developed calculus independently, though Leibniz's notation (dy/dx, ∫) proved more practical and is what's used today, while Newton's dot notation (ẋ) survives mainly in physics for time derivatives.
For functions of several variables, f(x,y), the partial derivative ∂f/∂x measures the rate of change with respect to x alone, holding y fixed. The gradient ∇f = (∂f/∂x, ∂f/∂y) is a vector pointing in the direction of steepest increase — fundamental to optimization algorithms including gradient descent, the workhorse of modern machine learning.
Weierstrass (1872) constructed a function that is continuous everywhere but differentiable nowhere — shocking contemporaries who assumed continuous functions must be differentiable "almost everywhere" at least. This example (an infinite sum of cosine waves at increasing frequencies) showed that continuity and differentiability are genuinely distinct properties, with continuity being the much weaker condition.
The concept generalises far beyond real-valued functions of a real variable. The Fréchet derivative extends differentiation to functions between Banach spaces (infinite-dimensional vector spaces), essential in the calculus of variations and functional analysis. In differential geometry, the derivative generalises to the covariant derivative, which accounts for curvature when differentiating vector fields on curved manifolds — a tool essential to general relativity.