Understanding the Derivative from First Principles (AP Calculus AB Unit 2)

Rate of change

A measure of how one quantity changes in response to another (often how f(x) changes as x changes).

Average rate of change

Change in output over change in input on an interval; (f(b)−f(a))/(b−a).

Secant line

The line through two points on a curve, (a,f(a)) and (b,f(b)); its slope equals the average rate of change on [a,b].

Secant slope

The slope of the secant line between $x=a$ and $x=b$; $\frac{(f(b)−f(a))}{(b−a)}$.

Instantaneous rate of change

How fast f(x) is changing at a specific x-value; the derivative at that point (slope of the tangent line).

Tangent line

The line that touches a curve at a point and matches its local direction there; its slope is the derivative at that point.

Difference quotient

An expression of the form (f(a+h)−f(a))/h (or equivalent) used to compute average rates of change and define derivatives.

Limit (in the context of derivatives)

A process describing what a quantity approaches as an input approaches a value (e.g., as $h→0$), used to define instantaneous rate of change without substituting $h=0$.

Derivative at a point

$f' (a) = \frac{lim(h→0) \big[f(a+h)−f(a)\big]}{h}$, if this limit exists; the tangent slope at $x=a$.

Derivative (informal meaning)

The formal name for instantaneous rate of change; also interpreted as the slope of the tangent line.

Derivative function

A function giving the derivative at each x where it exists: $f' (x) = \frac{lim(h→0)\big[f(x+h)−f(x)\big]}{h}$.

Equivalent limit definition of the derivative

$f' (a)=\frac{lim(x→a)\big[f(x)−f(a)\big]}{(x−a)}$, equivalent to the $h→0$ form via $x=a+h$.

Leibniz notation

The notation dy/dx for the derivative, read “dee y dee x,” meaning the instantaneous rate of change of y with respect to x.

Operator notation for derivatives

Notation such as d/dx (f(x)) indicating “take the derivative with respect to x.”

Average velocity

When f is position and x is time, (f(b)−f(a))/(b−a), measured in distance per time (e.g., m/s).

Instantaneous velocity

When $s(t)$ is position, $s' (t)$ gives velocity at an exact time $t$ (units: distance/time).

Right-hand (forward) difference quotient estimate

An estimate of $f' (a)$ using values to the right: $\frac{(f(a+h)−f(a))}{h}$ for small $h>0$.

Left-hand (backward) difference quotient estimate

An estimate of f′(a) using values to the left: [f(a)−f(a−h)]/h for small h>0.

Symmetric difference quotient

An often more accurate estimate of f′(a): [f(a+h)−f(a−h)]/(2h), using points equally spaced around a.

Differentiable at x=a

A function is differentiable at a if $f' (a)$ exists as a finite real number (the derivative limit exists).

Local linearity

The idea that if a function is differentiable at a point, then zooming in near that point makes the graph look more like a straight line.

Continuous at x=a

f is continuous at a if f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x)=f(a).

Differentiability implies continuity

Key fact: if f is differentiable at x=a, then f must be continuous at x=a (but not necessarily the other way around).

One-sided derivatives

Derivatives computed from the left and right: $\frac{lim(h→0−)\big[f(a+h)−f(a)\big]}{h}$ and $\frac{lim(h→0+)\big[f(a+h)−f(a)\big]}{h}$; both must agree for $f' (a)$ to exist.

Non-differentiable behaviors

Common reasons f′(a) fails to exist: corner (unequal finite one-sided slopes), cusp (slopes to opposite infinities), vertical tangent (unbounded slope), or discontinuity.