Derivative — Mathematics Knowledge System

Overview

The derivative measures how a function's output changes in response to an infinitesimally small change in its input — its instantaneous rate of change. Geometrically, it's the slope of the line tangent to the function's graph at a point. The derivative is the central object of differential calculus and underlies physics (velocity, acceleration), economics (marginal cost), and machine learning (gradient descent).

Intuition

Picture driving a car and watching the speedometer. Your average speed over a whole trip (distance ÷ time) can hide a lot — you might have been stopped at lights, then sped up on the highway. The derivative is what the speedometer shows at one exact instant: not an average over a stretch of time, but the rate of change AT that precise moment. Mathematically, you get there by computing the average rate of change over a shrinking interval, then asking what value that average approaches as the interval shrinks to a single point.

Interactive Graph

Formal Definition

Definition

The derivative of f at x is defined as the limit of the difference quotient (average rate of change over an interval) as that interval shrinks to zero:

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

The instantaneous rate of change of f at x

Definition

\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}

Leibniz notation, expressing the same limit as a ratio of infinitesimal changes

Notation

Notation	Meaning
$f'(x)$	Lagrange's notation — the derivative of f with respect to its variableAlso written: y'
$\frac{dy}{dx}$	Leibniz's notation — rate of change of y with respect to x
$\dot{x}$	Newton's dot notation — typically used for derivatives with respect to time
$Df(x)$	Operator notation — D as the differentiation operator
$f''(x)$	Second derivative — the derivative of the derivative (rate of change of the rate of change)

Derivation

Deriving the power rule (d/dx xⁿ = n·xⁿ⁻¹) directly from the limit definition, for the case f(x) = x²:

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

Start from the definition

= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}

Expand (x+h)²

= \lim_{h \to 0} \frac{2xh + h^2}{h}

The x² terms cancel

= \lim_{h \to 0} (2x + h)

Divide every term by h (valid since h ≠ 0 as we approach)

= 2x

Take the limit as h → 0

Proofs

Differentiability implies continuity

$\text{Assume } f'(a) \text{ exists.}$ (Given)
$f(x) - f(a) = \frac{f(x)-f(a)}{x-a} \cdot (x-a) \text{ for } x \neq a$ (Algebraic identity, valid since x≠a)
$\lim_{x\to a}[f(x)-f(a)] = \lim_{x\to a}\frac{f(x)-f(a)}{x-a} \cdot \lim_{x\to a}(x-a) = f'(a) \cdot 0 = 0$ (Product of limits; the difference quotient converges to f'(a) by hypothesis)
$\therefore \lim_{x\to a} f(x) = f(a)$ (Which is exactly the definition of continuity at a)

Properties

Power rule

\frac{d}{dx}x^n = nx^{n-1}

Sum rule

(f+g)' = f' + g'

Product rule

(fg)' = f'g + fg'

Quotient rule

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

Constant multiple rule

(cf)' = cf'

Linearity

\frac{d}{dx}[af(x)+bg(x)] = af'(x)+bg'(x)

Theorems

Theorem 1: Mean Value Theorem

If f is continuous on [a,b] and differentiable on (a,b), then there exists c \in (a,b) such that f'(c) = \frac{f(b)-f(a)}{b-a}.

Theorem 2: Rolle's Theorem

If f(a) = f(b) and f is continuous on [a,b], differentiable on (a,b), then there exists c \in (a,b) with f'(c) = 0.

Corollaries

Follows from Mean Value Theorem

If f'(x) = 0 everywhere on an interval, f is constant on that interval.

Applications

Velocity is the derivative of position with respect to time; acceleration is the derivative of velocity.

Formula Explorer

Animation

Animates a secant line through two points on the curve as the second point slides toward the first, visually converging into the tangent line — a direct visualization of the h → 0 limit in the definition.

Worked Examples

Apply the power rule to each term and the sum rule to combine them.
$\frac{d}{dx}[3x^2] = 6x,\quad \frac{d}{dx}[2x]=2,\quad \frac{d}{dx}[-5]=0$

Answer: f'(x) = 6x + 2

Practice Problems

Difficulty 3/10

Differentiate f(x) = 5x⁴ - 3x² + 7.

Difficulty 2/10

What is the geometric meaning of f'(a)?

The area under the curve up to x=aThe slope of the line tangent to f at x=aThe y-intercept of fThe average value of f on [0,a]

Difficulty 5/10

A ball's height is h(t) = -16t² + 64t + 5 (feet, seconds). Find its velocity at t=1.

Common Mistakes

Common Mistake

Applying the power rule to functions that aren't of the form xⁿ, e.g. treating d/dx[2ˣ] as x·2ˣ⁻¹.

The power rule only applies when the VARIABLE is the base and the exponent is constant. 2ˣ has a constant base and variable exponent — its derivative is 2ˣ ln(2), an entirely different rule (exponential functions).

Common Mistake

Forgetting the product rule and just multiplying the derivatives: (fg)' = f'g'.

This is false in general. The correct rule is (fg)' = f'g + fg'. Test it on f=g=x: (x·x)'=(x²)'=2x, but f'g'=1·1=1≠2x.

Common Mistake

Assuming a function is differentiable everywhere it's continuous.

Continuity does not imply differentiability. f(x)=|x| is continuous at x=0 but has a sharp corner — no single tangent line exists there.

Quiz

Flashcards

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Historical Background

Isaac Newton and Gottfried Wilhelm Leibniz independently developed the derivative in the 1660s–1670s while working on physics (Newton, motion and gravitation) and geometry (Leibniz, tangent lines), triggering a bitter priority dispute between England and continental Europe that lasted decades. Newton's 1666 'method of fluxions' treated the derivative as a rate of flow; Leibniz's 1675 notation (dy/dx) — built around infinitesimals — is the notation still used today. The concept wasn't placed on rigorous footing until Cauchy and Weierstrass formalized limits in the 19th century, retroactively justifying what Newton and Leibniz had done intuitively.

1665–1666
Newton develops the 'method of fluxions' during his plague-year isolation at Woolsthorpe
Isaac Newton
1675
Leibniz introduces the dy/dx notation and the term 'differential calculus'
Gottfried Wilhelm Leibniz
1704
Newton publishes his method publicly for the first time, in an appendix to Opticks
1823
Cauchy defines the derivative rigorously as a limit in Résumé des leçons
Augustin-Louis Cauchy

Summary

The derivative f'(x) measures the instantaneous rate of change of f at x.
Formally, f'(x) = lim(h→0) [f(x+h)-f(x)]/h — the limit of the average rate of change (slope of a secant line) as the interval shrinks to zero.
Geometrically, f'(a) is the slope of the tangent line at x=a.
Key rules: power, sum, product, quotient, constant multiple.
Differentiability implies continuity, but continuity does NOT imply differentiability (e.g. |x| at 0).

References

BookStewart, J. Calculus: Early Transcendentals, 8th ed. Ch. 3.
PaperNewton, I. (1666). De Analysi per Aequationes Numero Terminorum Infinitas (unpublished manuscript).
WebsiteWikipedia — Derivative