What is a derivative?

A derivative measures the instantaneous rate of change of a function at a point. It is defined as the limit of the difference quotient: f'(a) = lim(h→0) [f(a+h) - f(a)]/h. Geometrically, this equals the slope of the tangent line to the graph at that point.

What are the different notations for derivatives?

Four notations are common: Lagrange (f'(x)), Leibniz (dy/dx), Euler (Df), and Newton (dot notation). Lagrange is compact for functions, Leibniz behaves like a fraction in chain rule, Euler appears in operator theory, and Newton is used for time derivatives.

What does the derivative tell you about a function?

The sign of f'(x) indicates direction: positive means increasing, negative means decreasing. Where f'(x) = 0 are critical points (potential extrema). The second derivative f''(x) indicates concavity: positive is concave up, negative is concave down.

When is a function not differentiable?

A function fails to be differentiable at corners, cusps, vertical tangents, and discontinuities. Differentiability requires continuity (but continuity alone is not sufficient). The classic example is |x|, which is continuous but not differentiable at x = 0.

What are the basic differentiation rules?

The main rules are: power rule (d/dx[x^n] = nx^(n-1)), product rule, quotient rule, and chain rule for compositions. These convert the limit definition into algebraic procedures, making derivative computation efficient.

What is implicit differentiation?

Implicit differentiation finds dy/dx when y is not given explicitly as a function of x. Differentiate both sides of the equation with respect to x, treating y as a function of x (applying chain rule), then solve for dy/dx.

What are higher-order derivatives?

Higher-order derivatives result from differentiating repeatedly. The second derivative f''(x) measures concavity, the third and beyond capture finer behavior. Some functions exhibit patterns: e^x stays unchanged, polynomials eventually become zero, and trig functions cycle.

Derivatives

The Difference Quotient and Its Limit

Notation

Derivative Graph Analysis

The Derivative as a Function

Differentiability

Differentiation Rules

Differentiation Techniques

Derivatives of Common Functions

Derivatives of Special Functions

Higher-Order Derivatives

Differentials

Roadmap of the Derivatives Subtree

From Average to Instantaneous

The slope of a line measures constant rate of change. But most functions do not change at a constant rate—their graphs curve, steepen, and flatten. The derivative extends the idea of slope to curves, capturing how fast a function changes at a single point rather than over an interval.

The construction begins with the difference quotient: the slope of a secant line through two points on the graph. As the two points draw closer together, the secant line approaches the tangent line. The derivative is the limit of this process—the slope of the tangent line, defined precisely through a limiting operation.

This single concept—instantaneous rate of change via a limiting process—generates an entire framework. Rules convert the limit definition into efficient computation. Techniques extend differentiation to implicit and parametric settings. The derivative as a function reveals the shape of curves through sign analysis. Higher-order derivatives capture concavity and deeper layers of change.

Key Terms

The Derivative

Derivative— instantaneous rate of change:

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

Differentiability— the derivative exists and is finite

Tangent Line— best linear approximation at a point

Rates of Change

Instantaneous Rate of Change— rate at a single point, equal to

f'(a)

Average Rate of Change— slope of the secant line over an interval

Extensions

Higher-Order Derivative— derivatives of derivatives:

f'', f''', f^{(n)}

Differential—

dy = f'(x) \cdot dx

, the tangent-line estimate of change

Partial Derivative— derivative with respect to one variable, others held constant

Graph Analysis

Critical Point— where

f' = 0

f'

is undefined

Local Extremum— a local peak or valley

Concavity— bending direction, controlled by

f''

Inflection Point— where concavity reverses

Monotonic Function— strictly increasing or strictly decreasing

See All Calculus Definitions →

Notation

Several notation systems for derivatives coexist, each suited to different contexts.

Lagrange Notation

The symbol

f'(x)

denotes the derivative of

f

with respect to

x

. Higher derivatives use repeated primes:

f''(x)

f'''(x)

. For the

n

th derivative, parenthetical notation avoids stacking primes:

f^{(n)}(x)

. This notation is compact and emphasizes the function itself.

Leibniz Notation

The symbol

\frac{dy}{dx}

denotes the derivative of

y

with respect to

x

. It suggests a ratio of infinitesimal changes—and while the derivative is defined as a limit rather than a literal fraction, this notation behaves like a fraction in the chain rule and in differentials. Higher derivatives follow the pattern

\frac{d^2y}{dx^2}

\frac{d^3y}{dx^3}

. The operator form

\frac{d}{dx}[f(x)]

emphasizes differentiation as an operation applied to an expression.

Euler Notation

The operator

D

D_x

applied to a function:

Df(x)

means the same as

f'(x)

. Higher orders use powers of the operator:

D^2 f

D^n f

. This notation appears in differential equations and operator theory.

Newton Notation

A dot above the variable:

\dot{y}

for the first derivative,

\ddot{y}

for the second. Used almost exclusively where the independent variable is time. The notation signals that differentiation is with respect to

t

.

The table below collects the four systems side by side with their first-derivative and higher-derivative forms and the contexts where each one is most commonly used.

See All Calculus Symbols and Notations →

Notation	First derivative	Higher derivatives	Typical context
Lagrange	f'(x)	f''(x), f'''(x), f⁽ⁿ⁾(x)	compact, function-centric; standard in textbook calculus
Leibniz	dy ⁄ dx	d²y ⁄ dx², dⁿy ⁄ dxⁿ	behaves like a ratio; natural for chain rule and differentials
Euler	Df(x) or D_xf	D²f, Dⁿf	differential equations and operator theory
Newton	ẏ	ÿ, ⃛y	used almost exclusively when the independent variable is time

Derivative Graph Analysis

The sign of the derivative at a point reveals whether the function is increasing or decreasing there. Where

f'(x) > 0

, the function rises; where

f'(x) < 0

, the function falls. Points where

f'(x) = 0

f'(x)

is undefined are critical points—candidates for local maxima, minima, or neither.

The second derivative adds another layer. Where

f''(x) > 0

, the graph is concave up; where

f''(x) < 0

, the graph is concave down. Points where concavity changes are inflection points. Together, the first and second derivatives determine the complete shape of a curve: its direction, its turning points, and its bending.

Derivative graph analysis combines these tools into a systematic framework for understanding function behavior, from tangent line equations to optimization and curve sketching.

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The Derivative as a Function

The definition

f'(a)

produces a single number—the slope at one point. Replacing

a

with a variable

x

yields

f'(x)

, a function that assigns a slope to every point in its domain. The derivative function has its own graph, its own properties, and can itself be differentiated.

The graph of

f'

encodes the behavior of

f

: where

f

climbs,

f'

is positive; where

f

descends,

f'

is negative; where

f

has an extremum,

f'

crosses zero. Reading one graph from the other—and reversing the process—is a core skill.

The derivative as a function develops this perspective, including the domain of

f'

, the relationship between the graphs of

f

and

f'

, and the interpretation of

f'

as a standalone object.

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Differentiability

Not every function has a derivative at every point. The limit definition requires the limit to exist and be finite, which fails at corners, cusps, vertical tangents, and discontinuities. Each failure mode has a distinct geometric signature on the graph.

Differentiability implies continuity—a function must be unbroken at a point to have a tangent line there. The converse is false:

f(x) = |x|

is continuous at

x = 0

but not differentiable, because the left-hand and right-hand slopes disagree.

Differentiability examines when derivatives exist, one-sided derivatives, differentiability on intervals, and pathological cases where continuity holds everywhere but differentiability fails everywhere.

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Differentiation Rules

Computing derivatives from the limit definition is tedious for all but the simplest functions. Differentiation rules replace the limit with algebraic procedures: the power rule handles

x^n

, the product rule handles

f \cdot g

, the quotient rule handles

f/g

, and the chain rule handles compositions

f(g(x))

.

Beyond computational rules, the Mean Value Theorem guarantees that every differentiable function achieves its average rate of change at some interior point. Rolle's Theorem and L'Hôpital's rule follow as consequences, connecting derivatives to existence results and limit evaluation.

Differentiation rules covers all standard rules, their derivations from the limit definition, and the major theorems that govern derivative behavior.

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Differentiation Techniques

Standard rules apply directly when

y

is given explicitly as a function of

x

. Many relationships, however, are not written in explicit form. The equation

x^2 + y^2 = 25

defines

y

implicitly; implicit differentiation extracts

dy/dx

by differentiating both sides and solving.

Logarithmic differentiation handles products with many factors and expressions with variable exponents like

x^x

. Parametric differentiation computes slopes of curves given as

x = x(t)

y = y(t)

. The inverse function derivative formula recovers the derivative of

f^{-1}

from the derivative of

f

.

Differentiation techniques develops each method with its underlying logic and the situations where it applies.

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Derivatives of Common Functions

A small set of derivative formulas covers the vast majority of computation. The power rule gives

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

. Trigonometric derivatives follow a pattern:

(\sin x)' = \cos x

(\cos x)' = -\sin x

(\tan x)' = \sec^2 x

. The exponential

e^x

is its own derivative, and

(\ln x)' = 1/x

.

These formulas, combined with differentiation rules, handle polynomials, rational functions, and all standard transcendental functions. Memorizing them eliminates the need to return to the limit definition.

Derivatives of common functions catalogs these results, derives key formulas from the limit definition, and organizes them into a reference framework.

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Derivatives of Special Functions

Beyond the standard functions, several families have derivatives that appear frequently in advanced work. Inverse trigonometric functions produce algebraic derivatives:

(\arcsin x)' = 1/\sqrt{1 - x^2}

(\arctan x)' = 1/(1 + x^2)

. Hyperbolic functions parallel their trigonometric counterparts:

(\sinh x)' = \cosh x

(\tanh x)' = \text{sech}^2\,x

.

Piecewise functions require checking differentiability at each boundary, and parametric curves use the formula

dy/dx = (dy/dt)/(dx/dt)

.

Derivatives of special functions collects these results, derives them via implicit differentiation and inverse function methods, and highlights the patterns connecting them to common derivatives.

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Higher-Order Derivatives

Since

f'

is a function, it can be differentiated again. The second derivative

f''(x)

measures how the slope itself changes—geometrically, this is concavity. The third derivative and beyond capture progressively finer aspects of a function's behavior.

Certain functions exhibit patterns under repeated differentiation. Polynomials of degree

n

have

(n+1)

th derivative equal to zero. The exponential

e^x

is unchanged at every order. Sine and cosine cycle through four derivatives before repeating.

Higher-order derivatives explores notation, derivative patterns, smoothness classes, and the connection to Taylor series.

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Differentials

The derivative

dy/dx

is defined as a limit—a single object. Differentials separate it into two independent quantities:

dx

, a small change in

x

, and

dy = f'(x) \cdot dx

, the corresponding change predicted by the tangent line.

The differential

dy

approximates the actual change

\Delta y = f(x + dx) - f(x)

. For small

dx

, the approximation is accurate, making differentials a practical tool for linear approximation and error estimation.

Differentials develops the formalism, connects it to Leibniz notation, and applies it to approximation and error propagation.

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Roadmap of the Derivatives Subtree

The sections above describe each branch of the derivatives subtree in turn. The table below collects them as a single roadmap — grouping the nine sub-pages by category, naming what each one focuses on, and indicating the main payoff a reader should expect from following the link.

Sub-topic	Category	Focus	What it gives the reader
Graph analysis	Concept	signs of f' and f'' control function shape	direction, critical points, concavity, inflection
The derivative as a function	Concept	f'(x) as its own function with its own graph	reading f's behavior from f' and vice versa
Differentiability	Concept	when the derivative exists and when it fails	failure modes (corner, cusp, vertical tangent), continuity vs differentiability
Differentiation rules	Computation	algebraic procedures replacing the limit definition	power, product, quotient, chain rules; MVT, Rolle, L'Hôpital
Differentiation techniques	Computation	methods for non-explicit forms	implicit, logarithmic, parametric differentiation; inverse-function formula
Derivatives of common functions	Catalog	standard function derivatives as a reference set	power, trigonometric, exponential, and logarithmic formulas
Derivatives of special functions	Catalog	families beyond the standard set	inverse trigonometric, hyperbolic, piecewise, and parametric derivatives
Higher-order derivatives	Extension	repeated differentiation	concavity, derivative patterns, smoothness classes, connection to Taylor series
Differentials	Extension	dy = f'(x) · dx as a standalone quantity	linear approximation and error estimation