Unit 2: Differentiation: Definition and Fundamental Properties

Derivative

A measure of the instantaneous rate of change of a function; equivalently, the slope of the tangent line at a point (when it exists).

Average rate of change

The change in output divided by the change in input over an interval, given by (f(b)-f(a))/(b-a).

Difference quotient

An average-rate-of-change expression such as (f(b)-f(a))/(b-a) or (f(a+h)-f(a))/h.

Secant line

A line passing through two points on a curve, often (a,f(a)) and (b,f(b)).

Secant slope

The slope of a secant line, used to approximate a derivative; e.g., (f(a+h)-f(a))/h.

Instantaneous rate of change

How fast a quantity changes at a single input value; computed by the derivative via a limit.

Tangent line

A line that matches the curve’s direction (slope) at a point; it may intersect the curve again elsewhere.

Slope of the tangent line

The derivative value at that point, f'(a), if the derivative exists.

Limit definition of the derivative (h-form)

f'(a) = $\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$, if the limit exists and is finite.

Limit definition of the derivative (x→a form)

$f'(a) = \lim_{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$, an equivalent way to define the derivative at a.

Right-hand derivative

The derivative computed by approaching from the right: lim(h→0+) [f(a+h)-f(a)]/h.

Left-hand derivative

The derivative computed by approaching from the left: lim(h→0−) [f(a+h)-f(a)]/h.

Differentiability

A function is differentiable at a point if its derivative exists there (the defining limit exists and is finite).

Continuity

A function is continuous at a point if there is no hole/jump there and the function value matches the limiting value.

Differentiable implies continuous

Key fact: if f is differentiable at x=a, then f must be continuous at x=a (but not vice versa).

Corner

A point where the left-hand and right-hand slopes are finite but unequal, so the derivative does not exist.

Cusp

A sharp point where slopes become infinite in opposite directions, so the derivative does not exist.

Vertical tangent

A point where the slope becomes infinite (undefined), so the derivative does not exist there.

Discontinuity

A break in the function (jump, hole, or asymptote); differentiability fails at any discontinuity.

Symmetric difference quotient

A table-based estimate for f'(a): [f(a+h)-f(a-h)]/(2h), often more accurate than one-sided estimates.

One-sided secant estimate

An estimate of f'(a) using nearby data on one side: [f(a+h)-f(a)]/h.

Rise over run

Slope formula: (change in y)/(change in x); used for secant and tangent slopes.

Units of a derivative

Derivative units are (output units)/(input units), e.g., dollars per year or meters per second.

f'(x) (derivative function)

A new function that takes x-values and returns the slope/rate of change at each x.

f'(a) (derivative value at a point)

A single number: the slope/rate of change of f at the specific input x=a.

Derivative operator (d/dx)

Notation emphasizing differentiation as an operation: (d/dx)(f(x)).

dy/dx notation

Derivative of y with respect to x; commonly used in applied rate/units interpretations.

Second derivative

The derivative of the derivative, e.g., $f''(x)$ or $\frac{d^2y}{dx^2}$; measures how the first derivative changes.

Velocity

Instantaneous rate of change of position: v(t)=s'(t).

Acceleration

Instantaneous rate of change of velocity: $a(t) = v'(t) = s''(t)$.

Speed

The magnitude of velocity: speed = |v(t)|.

Marginal cost

In economics, the derivative of cost with respect to quantity; estimates the additional cost of producing one more item near a given output level.

Constant rule

The derivative of a constant is zero: d/dx(k)=0.

Constant multiple rule

A constant factor can be pulled out: d/dx(c f(x)) = c f'(x).

Sum rule (linearity)

Derivative of a sum is the sum of derivatives: d/dx(f+g)=f'+g'.

Difference rule (linearity)

Derivative of a difference is the difference of derivatives: $\frac{d}{dx}(f-g) = f' - g'$.

Power rule

For integer n, $\frac{d}{dx}(x^n) = n x^{n-1}$ (multiply by the power and subtract 1 from the exponent).

Negative exponent differentiation

Applying the power rule to $x^{-n}$ gives $\frac{d}{dx}(x^{-n}) = -n x^{-n-1}$, often rewritten with positive exponents.

Rewriting radicals and fractions using exponents

Converting forms like $\sqrt{x}$ and $\frac{1}{x^2}$ into $x^{1/2}$ and $x^{-2}$ to apply the power rule more easily.

Product rule

Derivative of a product: d/dx(fg)=f'g + f g' (not f'g').

Quotient rule

Derivative of a quotient: $\frac{d}{dx}(\frac{f}{g}) = \frac{f'g - fg'}{g^2}$ (with $g(x) \neq 0$).

“Low d-high minus high d-low”

Mnemonic for the quotient rule: denominator·(derivative of numerator) minus numerator·(derivative of denominator), over denominator squared.

Derivative of sin x

$\frac{d}{dx}(\sin \text{x}) = \cos \text{x}$.

Derivative of cos x

$\frac{d}{dx}(\cos \text{x}) = -\sin \text{x}$ (common sign mistake).

Derivative of tan x

$\frac{d}{dx}(\tan \text{x}) = \sec^2 \text{x}$ (where tan is defined).

Derivative of sec x

d/dx(sec x)=sec x·tan x (where sec is defined).

Derivative of e^x

d/dx(e^x)=e^x.

Derivative of ln x

$\frac{d}{dx}(\ln \text{x}) = \frac{1}{x}$ (for $x>0$ in real-valued contexts).

Conjugate (rationalizing technique)

An algebra tool (multiply by a conjugate) often used in limit-definition derivatives involving square roots to eliminate radicals.

Local linearity (zoom-in idea)

If a function is differentiable at a point, then when you zoom in near that point the graph looks more and more like a straight line.