Derivative from First Principles

The derivative from first principles computes the instantaneous rate of change of a function $f$ as the limit of average rates of change over shrinking intervals: $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$ . Key result: the slope of the tangent line at a point equals this limit, when it exists. Where it applies: any function differentiable at a point — used everywhere in calculus, physics, and optimisation. Who/when: formalised by Newton and Leibniz (1660s–1680s), made rigorous by Cauchy and Weierstrass (1800s).

The derivative from first principles defines the rate at which a function's output changes with respect to its input, at a single precise point. It is built directly from the limit definition, bypassing all shortcut rules, so it reveals why derivatives behave the way they do rather than just how to compute them. This matters because every differentiation rule — power, product, chain — is ultimately a theorem proved from this one definition. A surprising fact: even though the average rate of change $\frac{f ( x + h ) - f ( x )}{h}$ becomes the indeterminate form $\frac{0}{0}$ as $h \to 0$ , the limit can still settle on a perfectly definite finite value. This is the central magic of calculus: dividing two quantities that both vanish, yet getting a meaningful answer.

The Core Insight

A straight line has one slope everywhere; a curve does not. To assign a slope to a curve at a point, draw a secant line through two nearby points and let the second point slide toward the first. The secant's slope is something we can compute — just rise over run. As the gap $h$ shrinks to zero, the secant pivots into the tangent line, and its slope is the derivative. First principles means doing this limiting process explicitly, with no rules assumed.

Formal Statement

Derivative at a Point

Let $f$ be a function defined on an open interval containing $x$ . The derivative of $f$ at $x$ , denoted $f^{'} (x)$ , is

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

⟳

h

is the horizontal gap between the two points;

f (x + h) - f (x)

is the vertical gap (rise); their ratio is the secant slope (rise/run).

provided this limit exists and is finite. Here:

$x$ — the fixed point at which we measure the rate of change;
$h$ — the (signed) increment in the input, approaching $0$ from both sides;
$f (x + h) - f (x)$ — the change in output, $Δ y$ ;
$h$ — the change in input, $Δ x$ ;
$f^{'} (x)$ — the limiting slope, the derivative.

When the limit exists, $f$ is said to be differentiable at $x$ .

An equivalent form replaces $x + h$ with a roving point $a$ :

f^{'} (a) = x \to a lim \frac{f ( x ) - f ( a )}{x - a}

⟳substitute

x = a + h

, so

x \to a

matches

h \to 0

, and

x - a = h

; the two definitions are the same limit written differently.

Differentiability Implies Continuity

If $f$ is differentiable at $x = a$ , then $f$ is continuous at $x = a$ .

x \to a lim [f (x) - f (a)] = x \to a lim \frac{f ( x ) - f ( a )}{x - a} \cdot (x - a) = f^{'} (a) \cdot 0 = 0

⟳multiply and divide by

(x - a)

, then split the limit of a product into the product of limits since both factors converge.

The converse is false: continuity does not guarantee differentiability (see the counterexample below).

Why It Holds / Derivation

Derivative of $f(x)=x^2$ from First Principles

We compute $f^{'} (x)$ for $f (x) = x^{2}$ using only Equation [1].

Step 1 — Form the difference quotient. Substitute into the definition:

\frac{f ( x + h ) - f ( x )}{h} = \frac{( x + h ) ^{2} - x ^{2}}{h}

⟳directly plug

f (x + h) = (x + h)^{2}

and

f (x) = x^{2}

into the secant-slope ratio.

Step 2 — Expand the numerator.

(x + h)^{2} - x^{2} = x^{2} + 2 x h + h^{2} - x^{2} = 2 x h + h^{2}

⟳expand

(x + h)^{2} = x^{2} + 2 x h + h^{2}

and cancel the

x^{2}

terms.

Step 3 — Simplify the quotient by cancelling $h$ . For

h \neq = 0

\frac{2 x h + h ^{2}}{h} = 2 x + h

⟳factor

h

from the numerator, then cancel it with the denominator — legal because

h \neq = 0

throughout the limiting process.

Step 4 — Take the limit.

f^{'} (x) = h \to 0 lim (2 x + h) = 2 x

⟳the expression

2 x + h

is continuous in

h

, so substitute

h = 0

directly.

The crucial move is Step 3: cancelling $h$ removes the $\frac{0}{0}$ problem before we let $h \to 0$ . $■$

Isaac Newton | 1643–1727

Newton developed his "method of fluxions" around 1665–1666 during the plague years, treating quantities as flowing in time and derivatives ("fluxions") as their rates of flow. He wrote $\overset{x}{˙}$ for the fluxion of $x$ — a notation still used in physics for time derivatives. Newton used infinitesimally small increments he called "moments," denoted $o$ , and discarded higher-order terms in $o$ to obtain rates of change. Though his reasoning lacked the rigour of modern limits, it was operationally identical to first principles: form the increment, simplify, then let it vanish. His unpublished De Methodis (1671) and the Principia (1687) applied these ideas to motion, gravitation, and orbits.

Building Intuition

Imagine zooming in on a smooth curve at one point with an infinitely powerful microscope. The more you magnify, the straighter the curve looks — eventually indistinguishable from a straight line. That line is the tangent, and its slope is the derivative. This is local linearity: differentiable functions look linear up close.

Now think of the secant line through your point and a neighbour a distance $h$ away. When $h$ is large, the secant is a crude approximation of the local direction. As $h$ shrinks, the neighbour creeps closer and the secant aligns ever more snugly with the curve. The derivative is the slope the secant "wants to reach" as the neighbour merges with your point. We never actually set $h = 0$ (that would be $\frac{0}{0}$ ); we ask what value the slope approaches.

Secant lines converging to the tangent as $h\to 0$

Worked Examples

Velocity of a Falling Object

Given: A stone falls so its height drop is

s (t) = 5 t^{2}

metres after

t

seconds (taking

g \approx 10 m/s^{2}

). Find the instantaneous velocity at

t = 3

s using first principles. Step 1 — Difference quotient.

\frac{s ( 3 + h ) - s ( 3 )}{h} = \frac{5 ( 3 + h ) ^{2} - 5 ( 3 ) ^{2}}{h}

⟳substitute

t = 3

into the limit definition

s^{'} (t) = lim_{h \to 0} \frac{s ( t + h ) - s ( t )}{h}

Step 2 — Expand.

5 (9 + 6 h + h^{2}) - 45 = 45 + 30 h + 5 h^{2} - 45 = 30 h + 5 h^{2}

⟳expand

(3 + h)^{2} = 9 + 6 h + h^{2}

, distribute the

5

, cancel the constant

45

Step 3 — Cancel and take the limit.

s^{'} (3) = h \to 0 lim \frac{30 h + 5 h ^{2}}{h} = h \to 0 lim (30 + 5 h) = 30

⟳factor and cancel

h

, then set

h = 0

in the continuous remainder.

s^{'} (3) = 30 m/s

Derivative of the Reciprocal Function

Given:

f (x) = \frac{1}{x}

. Find

f^{'} (x)

for

x \neq = 0

from first principles. Step 1 — Difference quotient.

\frac{f ( x + h ) - f ( x )}{h} = \frac{\frac{1}{x + h} - \frac{1}{x}}{h}

⟳substitute

f (x + h) = \frac{1}{x + h}

and

f (x) = \frac{1}{x}

into the definition.

Step 2 — Combine the fractions in the numerator.

\frac{1}{x + h} - \frac{1}{x} = \frac{x - ( x + h )}{x ( x + h )} = \frac{- h}{x ( x + h )}

⟳use common denominator

x (x + h)

; the numerator

x - (x + h) = - h

Step 3 — Divide by $h$ and cancel.

\frac{1}{h} \cdot \frac{- h}{x ( x + h )} = \frac{- 1}{x ( x + h )}

⟳dividing by

h

is multiplying by

\frac{1}{h}

; the

h

in numerator cancels.

Step 4 — Take the limit.

f^{'} (x) = h \to 0 lim \frac{- 1}{x ( x + h )} = \frac{- 1}{x \cdot x} = - \frac{1}{x ^{2}}

⟳substitute

h = 0

in the denominator since it is continuous and nonzero for

x \neq = 0

f^{'} (x) = - \frac{1}{x ^{2}}

Common Misconceptions

You cannot just set $h = 0$ at the start. Doing so gives

\frac{f ( x ) - f ( x )}{0} = \frac{0}{0}

, which is undefined. The whole point of first principles is to algebraically simplify the difference quotient — cancelling the troublesome

h

— before taking the limit. Only after the

h

in the denominator is gone can you substitute

h = 0

safely. The derivative is not the value of the function.

f^{'} (x)

measures the slope, not the height, of

f

x

. A function can be large while its derivative is zero (a high flat plateau), or small while its derivative is huge (a steep climb near the origin).

Continuous but Not Differentiable: $f(x)=|x|$

The absolute value function $f (x) = ∣ x ∣$ is continuous everywhere, including at $x = 0$ , yet it has no derivative at $x = 0$ .

Approaching from the right ( $h > 0$ ): $\frac{∣0 + h ∣ - ∣0∣}{h} = \frac{h}{h} = 1$ .

Approaching from the left ( $h < 0$ ): $\frac{∣0 + h ∣ - ∣0∣}{h} = \frac{- h}{h} = - 1$ .

h \to 0^{+} lim \frac{∣ h ∣}{h} = 1 \neq = - 1 = h \to 0^{-} lim \frac{∣ h ∣}{h}

⟳for positive

h

∣ h ∣ = h

; for negative

h

∣ h ∣ = - h

; the one-sided limits disagree.

Because the left and right limits differ, the two-sided limit in Equation [1] does not exist. Geometrically, the graph has a sharp corner at the origin — there is no single tangent line. This proves that continuity is necessary but not sufficient for differentiability.

Connections

The derivative from first principles is the foundation on which all of differential calculus is built. The shortcut Power Rule, Product Rule, and Chain Rule are each theorems proved by applying Equation [1] to general forms — first principles is the engine, the rules are its compiled output. The definition relies entirely on the machinery of Limits, so every subtlety of limits (one-sided behaviour, existence, $\frac{0}{0}$ forms) transfers directly. Geometrically it produces the Tangent Line and underpins the Mean Value Theorem. Reversing the operation leads to the Antiderivative and ultimately to integration via the Fundamental Theorem of Calculus.

Real-World Applications

Kinematics (physics): velocity is the first-principles derivative of position with respect to time; acceleration is the derivative of velocity. The falling-stone example above is exactly this.
Marginal analysis (economics): marginal cost and marginal revenue are derivatives of total cost and revenue functions, telling firms the cost of producing one more unit.
Rates of reaction (chemistry): the instantaneous rate of a chemical reaction is the derivative of concentration with respect to time, found by shrinking the measurement interval.
Population dynamics (biology): the growth rate of a population is the derivative of population size over time, central to models like logistic growth.
Slope and optimisation (engineering): finding where $f^{'} (x) = 0$ locates maxima and minima — used to minimise material, energy, or cost.
Computer graphics: surface normals and lighting calculations use derivatives to find how steeply a surface tilts at each point.
Machine learning (gradient descent): training neural networks uses derivatives of a loss function to nudge parameters downhill toward better fits.

Historical Development

c. 250 BCE — Archimedes computes tangents and areas using the "method of exhaustion," a precursor to limiting arguments.
1629 — Pierre de Fermat finds maxima and minima by his method of "adequality," equating $f (x + e)$ and $f (x)$ then dividing by $e$ and setting $e = 0$ — strikingly close to first principles.
1665–1666 — Isaac Newton develops the method of fluxions during the plague years, treating motion and rates of flow.
1675–1684 — Gottfried Wilhelm Leibniz independently invents calculus with the $\frac{d y}{d x}$ notation and the differential $d x$ , publishing first in 1684.
1734 — George Berkeley publishes The Analyst, mocking infinitesimals as "ghosts of departed quantities" and exposing the logical gaps.
1755 — Leonhard Euler systematises differential calculus in Institutiones calculi differentialis.
1821 — Augustin-Louis Cauchy grounds the derivative rigorously in the limit of the difference quotient, in his Cours d'analyse.
1872 — Karl Weierstrass supplies the modern $ε$ – $δ$ definition of limits and exhibits a function continuous everywhere yet differentiable nowhere, settling the foundations.

Subtleties & Limits

First principles assumes the limit in Equation [1] exists — but functions can fail this in dramatic ways. Weierstrass's 1872 function is continuous at every point yet has no derivative anywhere; its graph is infinitely jagged at all scales (a fractal). This raises a deep question about how "typical" differentiable functions are: in a rigorous sense, almost all continuous functions are nowhere differentiable, making the smooth curves we draw the exceptional case rather than the rule.

A further subtlety concerns the gap between the intuitive "infinitesimal $h$ " used by Newton and Leibniz and the rigorous limit of Cauchy and Weierstrass. The 20th-century field of non-standard analysis (Abraham Robinson, 1960s) actually rehabilitated infinitesimals as legitimate numbers in an extended number system, showing that the original heuristic was not merely sloppy but could be made fully rigorous by a different route. Which foundation is "the right one" remains a matter of mathematical taste and pedagogy.

Derivative from First Principles

The Core Insight

Formal Statement

Derivative at a Point

Differentiability Implies Continuity

Why It Holds / Derivation

Derivative of $f(x)=x^2$ from First Principles

Isaac Newton | 1643–1727

Building Intuition

Secant lines converging to the tangent as $h\to 0$

Worked Examples

Velocity of a Falling Object

Derivative of the Reciprocal Function

Common Misconceptions

Continuous but Not Differentiable: $f(x)=|x|$

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads