Derivatives: Definition, Rules, Mean Value Theorem

⚡ Key Takeaways

The derivative measures the instantaneous rate of change of a function — the slope of its graph at a single point. It is defined as the limit of average rates of change as the interval shrinks to zero. Computed via a small set of rules (power, product, quotient, chain). The Mean Value Theorem guarantees that over any interval, the instantaneous rate equals the average rate somewhere inside. Foundations laid by Newton and Leibniz (1660s–1680s); MVT rigorized by Cauchy and Lagrange (1800s).

💡 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.

DefinitionThe Derivative

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

1

\quad f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

Derivation hint: take two points on the graph, $(a, f(a))$ and $(a+h, f(a+h))$ ; the slope of the secant line between them is rise over run, then let the run $h$ shrink to zero.

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

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

2

\quad f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}

Derivation hint: substitute $x = a + h$ into [1]; as $h \to 0$ , $x \to a$ . When $f'(a)$ exists for every $a$ in an interval, $f'$ is itself a function, also written $\frac{df}{dx}$ (Leibniz notation) or $\frac{dy}{dx}$ when $y = f(x)$ .

DefinitionDifferentiability 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).

TheoremDifferentiation Rules

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

3

\quad \frac{d}{dx}(x^n) = n\,x^{n-1}

Derivation hint: expand $(x+h)^n$ with the binomial theorem, subtract $x^n$ , divide by $h$ , and let $h \to 0$ ; only the $n x^{n-1}$ term survives.

4

\quad \frac{d}{dx}(cu) = c\,u', \qquad \frac{d}{dx}(u + v) = u' + v'

Derivation hint: substitute into [1]; limits distribute over sums and pull out constants.

5

\quad \frac{d}{dx}(uv) = u'v + uv' \quad\text{(Product Rule)}

Derivation hint: add and subtract $u(x+h)v(x)$ in the numerator of the difference quotient, then split into two limits.

6

\quad \frac{d}{dx}\!\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \quad\text{(Quotient Rule)}

Derivation hint: write $u/v$ and form the difference quotient over a common denominator $v(x)v(x+h)$ , then take the limit.

7

\quad \frac{d}{dx}\,f(g(x)) = f'(g(x))\cdot g'(x) \quad\text{(Chain Rule)}

Derivation hint: multiply and divide the difference quotient by $\Delta g = g(x+h)-g(x)$ , treating it as (rate of $f$ in $g$ ) $\times$ (rate of $g$ in $x$ ).

TheoremMean 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

8

\quad f'(c) = \frac{f(b) - f(a)}{b - a}

Derivation hint: the right side is the average slope; the theorem says the instantaneous slope hits this average somewhere — proved by tilting Rolle's theorem. The special case $f(a) = f(b)$ is Rolle's Theorem: there exists $c$ with $f'(c) = 0$ .

ProofMean Value Theorem from Rolle's Theorem

Step 1. First establish Rolle's Theorem. Suppose $f$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and $f(a)=f(b)$ . By the Extreme Value Theorem, $f$ attains a maximum and minimum on $[a,b]$ .

Step 2. If both extremes occur at the endpoints, then since $f(a)=f(b)$ , $f$ is constant and $f'(c)=0$ everywhere. Otherwise an extreme occurs at an interior point $c$ . At an interior maximum or minimum, $f'(c)=0$ (Fermat's stationary-point theorem: the difference quotient is $\le 0$ from one side and $\ge 0$ from the other, forcing the limit to be $0$ ).

Step 3. Now prove the full MVT. Define the auxiliary "tilt-correcting" function

9

\quad g(x) = f(x) - \left[\,f(a) + \frac{f(b)-f(a)}{b-a}\,(x - a)\right]

Derivation hint: subtract from $f$ the straight line (secant) joining the endpoints, so $g$ measures vertical gap from that chord.

Step 4. Check $g(a) = 0$ and $g(b) = 0$ by direct substitution. $g$ is continuous on $[a,b]$ and differentiable on $(a,b)$ since $f$ is and the subtracted line is.

Step 5. Apply Rolle's Theorem to $g$ : there is $c \in (a,b)$ with $g'(c)=0$ . Differentiating [9],

10

\quad g'(x) = f'(x) - \frac{f(b)-f(a)}{b-a}

Derivation hint: derivative of the linear term is its constant slope; the constant $f(a)$ vanishes.

Step 6. Setting $g'(c)=0$ gives $f'(c) = \dfrac{f(b)-f(a)}{b-a}$ , which is [8]. $\blacksquare$

∎

mathematician · 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.

🧠 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.

Figure

Secant slopes converging to the tangent at point

a

; the derivative is the limit

ExampleVelocity from a Position Function

Given: A ball's height (in metres) at time $t$ (seconds) is $s(t) = 20t - 5t^2$ . Find its velocity at $t = 1\text{ s}$ and the instant it is momentarily at rest.

Step 1. Velocity is the derivative of position. Apply [3] and [4] termwise:

11

\quad s'(t) = 20 - 10t

Derivation hint: derivative of $20t$ is $20$ ; derivative of $-5t^2$ is $-5\cdot 2t = -10t$ .

Step 2. Evaluate at $t = 1$ : $s'(1) = 20 - 10(1) = 10$ .

Step 3. The ball is at rest when $s'(t) = 0$ : $20 - 10t = 0 \Rightarrow t = 2$ .

Answer: Velocity at $t=1$ is $\boxed{10\ \text{m/s}}$ (upward); the ball is momentarily at rest at $\boxed{t = 2\ \text{s}}$ (the top of its flight).

ExampleChain and Product Rule Combined

Given: Differentiate $f(x) = x^2 \sqrt{3x + 1}$ .

Step 1. Write $\sqrt{3x+1} = (3x+1)^{1/2}$ . Identify a product $u = x^2$ , $v = (3x+1)^{1/2}$ .

Step 2. $u' = 2x$ (by [3]). For $v$ , use the chain rule [7] with outer power $1/2$ and inner $3x+1$ (derivative $3$ ):

12

\quad v' = \tfrac{1}{2}(3x+1)^{-1/2}\cdot 3 = \frac{3}{2\sqrt{3x+1}}

Derivation hint: bring down the exponent $1/2$ , reduce it by 1 to $-1/2$ , then multiply by the inner derivative $3$ .

Step 3. Apply the product rule [5]:

13

\quad f'(x) = 2x\sqrt{3x+1} + x^2\cdot\frac{3}{2\sqrt{3x+1}}

Derivation hint: $u'v + uv'$ with the pieces from Steps 1–2.

Step 4. Combine over a common denominator $2\sqrt{3x+1}$ :

14

\quad f'(x) = \frac{4x(3x+1) + 3x^2}{2\sqrt{3x+1}} = \frac{15x^2 + 4x}{2\sqrt{3x+1}}

Derivation hint: multiply first term top and bottom by $2\sqrt{3x+1}$ , expand $4x(3x+1)=12x^2+4x$ , add $3x^2$ .

Answer: $\boxed{f'(x) = \dfrac{15x^2 + 4x}{2\sqrt{3x+1}}}$

⚠️ Common mistake

"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.

CounterexampleContinuity 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 $-1$ :

15

\quad \lim_{h\to 0^+}\frac{|h|}{h} = +1, \qquad \lim_{h\to 0^-}\frac{|h|}{h} = -1

Derivation hint: for $h>0$ , $|h|=h$ so the quotient is $1$ ; for $h<0$ , $|h|=-h$ so the quotient is $-1$ . Since the two one-sided limits disagree, the limit in [1] does not exist, and $f'(0)$ is undefined — there is a corner, not a tangent line. (Weierstrass went further, constructing a function continuous everywhere but differentiable nowhere.)

connection

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 $f' > 0$ 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.

🔬 Where this appears

⏳ History

- 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'(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.

open-problem

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" $\frac{d^{1/2}f}{dx^{1/2}}$ should mean — an active research area with applications in viscoelasticity and anomalous diffusion.

Derivatives: Definition, Rules, Mean Value Theorem

Formal Statement

Why It Holds

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also