Derivative from First Principles

⚡ Key Takeaways

The derivative from first principles computes the instantaneous rate of change of a function ff as the limit of average rates of change over shrinking intervals: f′(x)=lim⁡h→0f(x+h)−f(x)hf'(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 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 hh 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

DefinitionDerivative at a Point

Let ff be a function defined on an open interval containing xx. The derivative of ff at xx, denoted f′(x)f'(x), is

1
f′(x)=lim⁡h→0f(x+h)−f(x)h\quad f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

Derivation hint: hh is the horizontal gap between the two points; f(x+h)−f(x)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:

  • xx — the fixed point at which we measure the rate of change;
  • hh — the (signed) increment in the input, approaching 00 from both sides;
  • f(x+h)−f(x)f(x+h)-f(x) — the change in output, Δy\Delta y;
  • hh — the change in input, Δx\Delta x;
  • f′(x)f'(x) — the limiting slope, the derivative.

When the limit exists, ff is said to be differentiable at xx.

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

2
f′(a)=lim⁡x→af(x)−f(a)x−a\quad f'(a) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}

Derivation hint: substitute x=a+hx = a+h, so x→ax\to a matches h→0h\to 0, and x−a=hx-a = h; the two definitions are the same limit written differently.

TheoremDifferentiability Implies Continuity

If ff is differentiable at x=ax = a, then ff is continuous at x=ax = a.

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lim⁡x→a[f(x)−f(a)]=lim⁡x→af(x)−f(a)x−a⋅(x−a)=f′(a)⋅0=0\quad \lim_{x\to a}\big[f(x)-f(a)\big] = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\cdot(x-a) = f'(a)\cdot 0 = 0

Derivation hint: multiply and divide by (x−a)(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

ProofDerivative of f(x)=x2f(x)=x^2 from First Principles

We compute f′(x)f'(x) for f(x)=x2f(x)=x^2 using only Equation [1].

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

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f(x+h)−f(x)h=(x+h)2−x2h\quad \frac{f(x+h)-f(x)}{h} = \frac{(x+h)^2 - x^2}{h}

Derivation hint: directly plug f(x+h)=(x+h)2f(x+h)=(x+h)^2 and f(x)=x2f(x)=x^2 into the secant-slope ratio.

Step 2 — Expand the numerator.

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(x+h)2−x2=x2+2xh+h2−x2=2xh+h2\quad (x+h)^2 - x^2 = x^2 + 2xh + h^2 - x^2 = 2xh + h^2

Derivation hint: expand (x+h)2=x2+2xh+h2(x+h)^2 = x^2+2xh+h^2 and cancel the x2x^2 terms.

Step 3 — Simplify the quotient by cancelling hh. For h≠0h\ne 0:

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2xh+h2h=2x+h\quad \frac{2xh + h^2}{h} = 2x + h

Derivation hint: factor hh from the numerator, then cancel it with the denominator — legal because h≠0h\ne 0 throughout the limiting process.

Step 4 — Take the limit.

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f′(x)=lim⁡h→0(2x+h)=2x\quad f'(x) = \lim_{h\to 0}(2x + h) = 2x

Derivation hint: the expression 2x+h2x+h is continuous in hh, so substitute h=0h=0 directly. The crucial move is Step 3: cancelling hh removes the 00\tfrac{0}{0} problem before we let h→0h\to 0. ■\blacksquare

∎
IN
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 x˙\dot{x} for the fluxion of xx — a notation still used in physics for time derivatives. Newton used infinitesimally small increments he called "moments," denoted oo, and discarded higher-order terms in oo 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

🧠 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 hh away. When hh is large, the secant is a crude approximation of the local direction. As hh 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=0h=0 (that would be 00\tfrac{0}{0}); we ask what value the slope approaches.

Figure
Secant lines converging to the tangent as h→0h\to 0
x y f(x) (x, f(x)) (x+h, f(x+h)) secant (large h) tangent (h→0) h

Worked Examples

ExampleVelocity of a Falling Object

Given: A stone falls so its height drop is s(t)=5t2s(t) = 5t^2 metres after tt seconds (taking g≈10 m/s2g\approx 10\ \text{m/s}^2). Find the instantaneous velocity at t=3t = 3 s using first principles.

Step 1 — Difference quotient.

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

Derivation hint: substitute t=3t=3 into the limit definition s′(t)=lim⁡h→0s(t+h)−s(t)hs'(t)=\lim_{h\to0}\frac{s(t+h)-s(t)}{h}.

Step 2 — Expand.

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5(9+6h+h2)−45=45+30h+5h2−45=30h+5h2\quad 5(9 + 6h + h^2) - 45 = 45 + 30h + 5h^2 - 45 = 30h + 5h^2

Derivation hint: expand (3+h)2=9+6h+h2(3+h)^2=9+6h+h^2, distribute the 55, cancel the constant 4545.

Step 3 — Cancel and take the limit.

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s′(3)=lim⁡h→030h+5h2h=lim⁡h→0(30+5h)=30\quad s'(3) = \lim_{h\to 0}\frac{30h + 5h^2}{h} = \lim_{h\to 0}(30 + 5h) = 30

Derivation hint: factor and cancel hh, then set h=0h=0 in the continuous remainder.

s′(3)=30 m/s\boxed{s'(3) = 30\ \text{m/s}}
ExampleDerivative of the Reciprocal Function

Given: f(x)=1xf(x) = \dfrac{1}{x}. Find f′(x)f'(x) for x≠0x\ne 0 from first principles.

Step 1 — Difference quotient.

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

Derivation hint: substitute f(x+h)=1x+hf(x+h)=\frac{1}{x+h} and f(x)=1xf(x)=\frac{1}{x} into the definition.

Step 2 — Combine the fractions in the numerator.

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

Derivation hint: use common denominator x(x+h)x(x+h); the numerator x−(x+h)=−hx-(x+h) = -h.

Step 3 — Divide by hh and cancel.

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1h⋅−hx(x+h)=−1x(x+h)\quad \frac{1}{h}\cdot\frac{-h}{x(x+h)} = \frac{-1}{x(x+h)}

Derivation hint: dividing by hh is multiplying by 1h\tfrac{1}{h}; the hh in numerator cancels.

Step 4 — Take the limit.

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f′(x)=lim⁡h→0−1x(x+h)=−1x⋅x=−1x2\quad f'(x) = \lim_{h\to 0}\frac{-1}{x(x+h)} = \frac{-1}{x\cdot x} = -\frac{1}{x^2}

Derivation hint: substitute h=0h=0 in the denominator since it is continuous and nonzero for x≠0x\ne 0.

f′(x)=−1x2\boxed{f'(x) = -\dfrac{1}{x^2}}

Common Misconceptions

⚠️ Common mistake

You cannot just set h=0h = 0 at the start. Doing so gives f(x)−f(x)0=00\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 hh — before taking the limit. Only after the hh in the denominator is gone can you substitute h=0h=0 safely.

The derivative is not the value of the function. f′(x)f'(x) measures the slope, not the height, of ff at xx. 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).

CounterexampleContinuous but Not Differentiable: f(x)=∣x∣f(x)=|x|

The absolute value function f(x)=∣x∣f(x)=|x| is continuous everywhere, including at x=0x=0, yet it has no derivative at x=0x=0. Approaching from the right (h>0h>0): ∣0+h∣−∣0∣h=hh=1\dfrac{|0+h|-|0|}{h} = \dfrac{h}{h} = 1. Approaching from the left (h<0h<0): ∣0+h∣−∣0∣h=−hh=−1\dfrac{|0+h|-|0|}{h} = \dfrac{-h}{h} = -1.

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lim⁡h→0+∣h∣h=1≠−1=lim⁡h→0−∣h∣h\quad \lim_{h\to 0^+}\frac{|h|}{h} = 1 \ne -1 = \lim_{h\to 0^-}\frac{|h|}{h}

Derivation hint: for positive hh, ∣h∣=h|h|=h; for negative hh, ∣h∣=−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

connection

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, 00\tfrac{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

🔬 Where this appears

Historical Development

⏳ History
- 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)f(x+e) and f(x)f(x) then dividing by ee and setting e=0e=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 dydx\frac{dy}{dx} notation and the differential dxdx, 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\varepsilon–δ\delta definition of limits and exhibits a function continuous everywhere yet differentiable nowhere, settling the foundations.

Subtleties & Limits

open-problem

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 hh" 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.

See Also

Limits · Tangent Line · Power Rule · Continuity · Mean Value Theorem · Implicit Differentiation · Fundamental Theorem of Calculus · Antiderivative