Derivatives: Definition, Rules, Mean Value Theorem

The derivative of a function $f$ at a point $a$, written $f^{\prime }(a)$, is the limit of the difference quotient $\frac{f(a+h)-f(a)}{h}$ as $h\to 0$. Geometrically it is the slope of the tangent line; physically it is an instantaneous rate (velocity from position, current from charge). It matters because almost every quantitative science is written in the language of rates of change, and the derivative is the precise tool for capturing them. A surprising fact: a function can be continuous everywhere yet differentiable nowhere — smoothness is a strictly stronger condition than connectedness. The Mean Value Theorem ties the local notion (slope at a point) to global behavior (average slope over an interval), and underlies nearly every theoretical result in calculus.

The Core Insight

You cannot find a "slope at one point" by the usual rise-over-run formula — that needs two points. The trick is to compute the slope between two nearby points and then watch what happens as they merge. The derivative is the limit of that process. Differentiation rules are simply shortcuts that let you avoid recomputing this limit every time.

Formal Statement

The Derivative

The derivative of a function $f$ at a point $a$ is

• $f$ — the function being differentiated
• $a$ — the point at which the rate is measured
• $h$ — the (signed) horizontal gap between the two points
• $f^{\prime }(a)$ — the resulting slope; if the limit exists, $f$ is differentiable at $a$

An equivalent form, putting $x=a+h$, is

When $f^{\prime }(a)$ exists for every $a$ in an interval, $f^{\prime }$ is itself a function, also written $\frac{df}{dx}$ (Leibniz notation) or $\frac{dy}{dx}$ when $y=f(x)$.

Differentiability Implies Continuity

If $f$ is differentiable at $a$, then $f$ is continuous at $a$. The converse is false (see Continuity of Functions and the counterexample below).

Differentiation Rules

For differentiable functions $u(x)$ and $v(x)$ and constants $c,n$:

Mean Value Theorem

If $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists at least one point $c\in (a,b)$ such that

The special case $f(a)=f(b)$ is Rolle's Theorem: there exists $c$ with $f^{\prime }(c)=0$.

Why It Holds

Mean Value Theorem from Rolle's Theorem

Gottfried Wilhelm Leibniz | 1646–1716

A German polymath — philosopher, diplomat, and mathematician — who independently developed calculus in the 1670s–80s, slightly later than Newton but publishing first (1684). Leibniz gave us the notation we still use: $dx$, $dy$, $\frac{dy}{dx}$, and the integral sign $\int$. His symbolism treated derivatives as ratios of infinitesimals, making the chain rule look like algebraic cancellation. A bitter priority dispute with Newton's supporters dominated his later years, but Leibniz's superior notation is why Continental calculus flourished. He also invented binary arithmetic and built a mechanical calculator.

Building Intuition

Think of driving a car. Your position is a function of time. The average velocity over a trip is total distance divided by total time — that's the secant slope, $\frac{f(b)-f(a)}{b-a}$. Your instantaneous velocity is what the speedometer reads at one instant — that's the derivative, the limit as the time window shrinks to zero. The Mean Value Theorem then says something obvious in hindsight: if your average speed over a trip was 60 mph, then at some instant your speedometer read exactly 60 mph. You couldn't have been always above or always below the average. This is also why speed cameras on toll roads work: two timestamps give the average speed, and the MVT proves you must have hit that speed somewhere.

Secant slopes converging to the tangent at point $a$; the derivative is the limit

Worked Examples

Velocity from a Position Function

Chain and Product Rule Combined

Common Misconceptions

• "Continuous means differentiable." False. A graph can be unbroken yet have a sharp corner (like $|x|$ at $0$) where no single tangent slope exists.
• The MVT does not say $c$ is unique — there may be several valid points $c$. It only guarantees at least one.
• $\frac{dy}{dx}$ is not a fraction. It behaves like one in the chain rule for memory purposes, but it is the limit symbol of [1], not literally $dy$ divided by $dx$.
• Forgetting the inner derivative in the chain rule is the single most common error: $\frac{d}{dx}\sin (3x)=3\cos (3x)$, not $\cos (3x)$.
• The MVT requires continuity on the closed interval and differentiability on the open interval. Drop either hypothesis and the conclusion can fail.

Continuity Does Not Imply Differentiability

The absolute value function $f(x)=|x|$ is continuous everywhere, including at $x=0$. But the difference quotient from the right gives slope $+1$, while from the left it gives slope $-1$:

Since the two one-sided limits disagree, the limit in [1] does not exist, and $f^{\prime }(0)$ is undefined — there is a corner, not a tangent line. (Weierstrass went further, constructing a function continuous everywhere but differentiable nowhere.)

Connections

The derivative is one half of calculus; its inverse operation is the Definite and Indefinite Integrals, joined by the Fundamental Theorem of Calculus. The MVT is the engine behind Increasing and Decreasing Functions (if $f^{\prime }>0$ on an interval, $f$ is increasing there) and underlies L'Hôpital's Rule for indeterminate limits. The definition rests entirely on Limits of Functions and Continuity of Functions. Second derivatives extend the idea to Concavity and Inflection Points, and the whole framework generalizes to multiple variables through Partial Derivatives.

Real-World Applications

1. Physics — kinematics: velocity is the derivative of position; acceleration is the derivative of velocity ($v=s^{\prime }$, $a=v^{\prime }$).
2. Economics — marginal analysis: marginal cost and marginal revenue are derivatives of total cost and revenue functions; profit is maximized where $C^{\prime }(q)=R^{\prime }(q)$.
3. Engineering — optimization: finding the dimensions that minimize material or maximize strength sets $f^{\prime }(x)=0$ (critical points).
4. Medicine — pharmacokinetics: the rate at which drug concentration changes in the bloodstream is a derivative used to set dosing intervals.
5. Average-speed enforcement (SPECS cameras): two timed photographs give average speed; the MVT proves the driver must have momentarily reached that speed, making the ticket logically airtight.
6. Machine learning — gradient descent: training neural networks minimizes error by repeatedly stepping in the direction of the negative derivative (gradient).
7. Climate and population modeling: growth rates ($\frac{dP}{dt}$) drive differential-equation models of populations, CO₂ levels, and epidemics.

Historical Development

• c. 1629 — Pierre de Fermat develops a method of "adequality" to find maxima, minima, and tangents, a precursor to setting the derivative to zero.
• 1665–1666 — Isaac Newton, during plague-year isolation, develops "fluxions" — rates of change of "fluents" (flowing quantities) — and the basic rules.
• 1675–1684 — Gottfried Leibniz independently invents calculus with the $dx$, $dy$ notation; publishes the differential calculus in 1684 in Acta Eruditorum.
• 1691 — Michel Rolle proves the special case now called Rolle's Theorem (initially as an algebra result, ironically skeptical of calculus).
• 1715 — Brook Taylor publishes Taylor series, expressing functions via their derivatives.
• 1797 — Joseph-Louis Lagrange introduces the prime notation $f^{\prime }(x)$ and tries to ground calculus in algebra, stating an early Mean Value Theorem.
• 1821 — Augustin-Louis Cauchy gives the first rigorous limit-based definition of the derivative and a rigorous MVT in his Cours d'analyse.
• 1872 — Karl Weierstrass constructs a function that is continuous everywhere but differentiable nowhere, shattering the assumption that continuity implies smoothness.

Subtleties & Limits

The derivative as defined here is pointwise and requires a two-sided limit to exist. But many natural functions — corners, cusps, fractals — defy it. Several deep questions and extensions arise:

• Non-differentiable continuity: Weierstrass's nowhere-differentiable function shows the "typical" continuous function is in fact non-smooth almost everywhere — a fact made precise in measure theory.
• Weak derivatives: modern analysis (Sobolev spaces, distributions) defines generalized derivatives for functions too rough for [1], essential for partial differential equations.
• The MVT fails for vector-valued and complex functions in its equality form; only inequality versions survive, a subtlety often glossed over.
• Fractional calculus asks what a "half-th derivative" $\frac{d^{1/2}f}{d{x}^{1/2}}$ should mean — an active research area with applications in viscoelasticity and anomalous diffusion.

See Also