Unit 2: Differentiation: Definition and Fundamental Properties

Average rate of change

The change in function output per unit change in input over an interval; (f(b)−f(a))/(b−a).

Average velocity

For position f(x) over time x, the average velocity from x=a to x=b is (f(b)−f(a))/(b−a).

Secant line

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

Rise over run

Coordinate form of slope: $\frac{y_2-y_1}{x_2-x_1}$.

Instantaneous rate of change

The rate of change at a single input value (an “instant”); computed using the derivative (a limit).

Difference quotient

The secant slope using a+h and a: (f(a+h)−f(a))/h.

Tangent line

The limiting position of secant lines as the two points merge; its slope is the derivative at that point.

Derivative at a point

$f′(a)=\lim_{h \to 0} \frac{(f(a+h)-f(a))}{h},$ if this limit exists and is finite.

Alternate limit definition of derivative

$$f′(a)=\lim_{x \to a} \frac{(f(x)-f(a))}{(x-a)}.$$

Limit (in the derivative definition)

The process of letting h→0 (or x→a) to capture the instantaneous slope/rate of change.

Critical algebra step (canceling h)

In difference quotients, factor to cancel a common factor of h before taking the limit (since h=0 is not allowed in the quotient).

Early substitution error

The common mistake of plugging in h=0 before simplifying (the quotient is undefined at h=0).

Derivative value $f'(2)$ for $f(x)=x^2$

Using the limit definition gives $f′(2)=4,$ the tangent slope at $x=2.$

Derivative as a function

f′(x) is a new function giving the slope of the tangent line to f at each x.

First-principles derivative of $x^2$

Using the limit definition, if $f(x)=x^2$ then $f'(x)=2x$.

Notation: f′(x)

A common notation meaning “the derivative of f with respect to x.”

Notation: y′

Derivative notation when y is used for the output variable; equivalent to dy/dx.

Notation: dy/dx

Read “derivative of y with respect to x”; emphasizes rate and units.

Notation: d/dx(f(x))

Operator notation meaning “differentiate f(x) with respect to x.”

Derivative evaluated at a point

Notation such as f′(a) or (dy/dx)|_{x=a} meaning the derivative value at x=a.

Second derivative

The derivative of the derivative (e.g., f′′(x)); measures how f′ is changing.

Increasing on an interval

If $f'(x) > 0$ on an interval, then $f(x)$ is increasing on that interval.

Decreasing on an interval

If $f′(x)<0$ on an interval, then $f(x)$ is decreasing on that interval.

Horizontal tangent

A tangent line with slope $0$; occurs where $f′(a)=0$ (does not automatically imply a max/min).

Derivative as slope

Geometric interpretation: f′(a) is the slope of the tangent line to y=f(x) at x=a.

Derivative units rule

If f has units and x has units, then f′ has units “(units of f) per (unit of x).”

Velocity

If s(t) is position, velocity is v(t)=s′(t).

Acceleration

If v(t) is velocity, acceleration is a(t)=v′(t)=s′′(t).

Linearization (tangent line approximation)

Near x=a, f(x) is approximated by L(x)=f(a)+f′(a)(x−a).

Tangent line formula

The tangent line at x=a: y = f(a) + f′(a)(x−a) (equivalently point-slope form through (a,f(a))).

Differentiable at x=a

A function is differentiable at a if the derivative f′(a) exists (the defining limit exists and is finite).

Continuous at x=a

A function is continuous at a if lim(x→a) f(x) = f(a).

Differentiability implies continuity

If f is differentiable at a, then f is continuous at a (but not conversely).

Corner (non-differentiability)

A point where left-hand and right-hand slopes are finite but unequal, so f′(a) does not exist.

Cusp (non-differentiability)

A point where slopes become infinite in opposite directions, so the derivative does not exist as a finite number.

Vertical tangent (non-differentiability)

A point where slope becomes infinite; the derivative does not exist as a finite value.

Discontinuity (blocks differentiability)

If a function has a jump, hole, or blow-up at a, it cannot be differentiable at a.

One-sided derivative test

Compare $\lim_{h\to0^{-}} \frac{(f(a+h)-f(a))}{h}$ and $\lim_{h\to0^{+}} \frac{(f(a+h)-f(a))}{h}$; if not equal/finite, $f′(a)$ does not exist.

Absolute value corner example

For f(x)=|x| at x=0, the difference quotient approaches −1 from the left and 1 from the right, so f′(0) does not exist.

Piecewise continuity condition at a join

To be continuous at x=a: lim(x→a−)f(x)=lim(x→a+)f(x)=f(a).

Piecewise differentiability condition at a join

To be differentiable at x=a: f must be continuous at a and the one-sided derivatives must match.

Constant rule

d/dx(k)=0 for a constant k.

Constant multiple rule

d/dx(kf(x)) = kf′(x).

Sum and difference rules

d/dx(f+g)=f′+g′ and d/dx(f−g)=f′−g′.

Power rule

For integer $n$ (on its domain), $\frac{d}{dx}(x^n)=n x^{n-1}.$

Product rule

d/dx(uv)=u(dv/dx)+v(du/dx).

Quotient rule

$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v \frac{du}{dx}-u \frac{dv}{dx}}{v^2}$ (with $v\neq0$).

Derivative of e^x

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

Derivative of a^x

For a > 0, a ≠ 1: $\frac{d}{dx}(a^x)=a^x \ln(a).$

Derivative of ln(x)

For $x > 0$: $\frac{d}{dx}(\ln(x))=\frac{1}{x}.$

Core trig derivatives (radians)

$\frac{d}{dx}(\sin x)=\cos x, \frac{d}{dx}(\cos x)=-\sin x,$ and $\frac{d}{dx}(\tan x)=\sec^2 x$ (assuming $x$ is in radians).