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: (y2−y1)/(x2−x1).

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→0) (f(a+h)−f(a))/h, if this limit exists and is finite.

Alternate limit definition of derivative

f′(a)=lim(x→a) (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→0−) (f(a+h)−f(a))/h and lim(h→0+) (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), d/dx(x^n)=n x^{n−1}.

Product rule

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

Quotient rule

d/dx(u/v)=(v du/dx − u dv/dx)/v^2 (with v≠0).

Derivative of e^x

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

Derivative of a^x

For a>0, a≠1: d/dx(a^x)=a^x ln(a).

Derivative of ln(x)

For x>0: d/dx(ln(x))=1/x.

Core trig derivatives (radians)

d/dx(sin x)=cos x, d/dx(cos x)=−sin x, and d/dx(tan x)=sec^2 x (assuming x is in radians).