Unit 2: Differentiation: Definition and Fundamental Properties

Average rate of change

The change in a function over an interval: (f(b)−f(a))/(b−a); interpreted as average “output per input” from x=a to x=b.

Secant line

A line that intersects a curve at two points; used to model average change over an interval.

Slope (rise over run)

The rate of change of a line: (y2−y1)/(x2−x1); measures vertical change per horizontal change.

Instantaneous rate of change

How fast a quantity is changing at a single input value (the “right now” rate); equals the slope of the tangent line at that point.

Tangent line

The line that best matches a curve at a single point; its slope represents the instantaneous rate of change there.

Difference quotient

An expression like (f(a+h)−f(a))/h that computes a secant slope; its limit as h→0 defines the derivative.

Limit (in derivative context)

A value that an expression approaches as the input approaches a certain number; used to make “instantaneous” change precise.

Increment h

A small change in the input x (moving from a to a+h); in derivatives, h is taken toward 0.

Units of a rate

Derivative/average rate units are (output units)/(input units), e.g., meters per second if f is meters and x is seconds.

Average velocity

Average rate of change of position s(t) over a time interval: (s(t2)−s(t1))/(t2−t1).

Forward difference quotient

A right-hand secant estimate of f'(a): (f(a+h)−f(a))/h for small h>0.

Backward difference quotient

A left-hand secant estimate of f'(a): (f(a)−f(a−h))/h for small h>0.

Symmetric difference quotient

A centered estimate of f'(a): (f(a+h)−f(a−h))/(2h); often more accurate because it uses both sides.

Derivative

A limit that gives the instantaneous rate of change (tangent slope) of a function at a point; written f'(a) or dy/dx.

Limit definition of the derivative (h→0 form)

f'(a)=lim_{h→0} (f(a+h)−f(a))/h, if the limit exists as a finite real number.

Equivalent derivative definition (x→a form)

f'(a)=lim_{x→a} (f(x)−f(a))/(x−a); same concept as the h→0 form.

Derivative exists

The derivative at x=a exists if the difference-quotient limit exists and is a finite real number (left and right behaviors agree).

One-sided derivative

A derivative computed using a limit from only one side (left or right) of the point.

Left-hand derivative

lim_{h→0−} (f(a+h)−f(a))/h; uses values approaching a from the left.

Right-hand derivative

lim_{h→0+} (f(a+h)−f(a))/h; uses values approaching a from the right.

Derivative function

The function f'(x) that gives the slope (instantaneous rate of change) of f at each x where the derivative exists.

f'(a)

Notation for the derivative evaluated at x=a; a single number representing the tangent slope at that point.

f'(x)

Notation for the derivative as a function of x; outputs the slope of f at each input where defined.

Leibniz notation (dy/dx)

A common derivative notation emphasizing “rate of change of y with respect to x”; conceptually defined by a limit.

(dy/dx)|_{x=a}

Leibniz notation for the derivative evaluated at a specific point x=a (the rate/slope at that input).

Second derivative

The derivative of the derivative, f''(x) or d^2y/dx^2; measures how the rate of change itself is changing.

Differentiable at a point

A function is differentiable at x=a if f'(a) exists (the tangent slope is well-defined and finite).

Differentiability implies continuity

If f is differentiable at x=a, then f is continuous at x=a (but continuous does not always imply differentiable).

Continuous but not differentiable

A situation where a graph has no break at a point but still has no well-defined tangent slope there (e.g., a corner).

Discontinuity

A break in the graph (hole, jump, vertical asymptote, etc.); prevents differentiability at that x-value.

Corner

A sharp turn where left-hand and right-hand slopes are finite but unequal (e.g., |x| at 0); derivative does not exist there.

Cusp

A pointy feature where slopes become infinite in different ways; derivative does not exist as a finite real number.

Vertical tangent

A tangent line with infinite/undefined slope; derivative does not exist as a finite real number at that point.

Constant rule

d/dx(c)=0 for any constant c; a horizontal line has slope 0 everywhere.

Constant multiple rule

d/dx(kf(x))=k f'(x); multiplying a function by k multiplies its derivative by k.

Power rule

For integer n>0, d/dx(x^n)=n x^(n−1) (multiply by the power and subtract 1 from the exponent).

Sum and difference rules

d/dx(f+g)=f'+g' and d/dx(f−g)=f'−g'; derivatives distribute over addition/subtraction.

Polynomial differentiation

Differentiating a polynomial term-by-term using the power rule plus sum/difference and constant multiple rules.

Point-slope form

A line through (a, f(a)) with slope m: y−f(a)=m(x−a).

Tangent line equation

The line tangent to y=f(x) at x=a: y−f(a)=f'(a)(x−a).

Product rule

If h(x)=f(x)g(x), then h'(x)=f'(x)g(x)+f(x)g'(x) (not f'(x)g'(x)).

Quotient rule

If h(x)=f(x)/g(x) with g(x)≠0, then h'(x)=[f'(x)g(x)−f(x)g'(x)]/[g(x)]^2.

“Low d-high minus high d-low”

Mnemonic for the quotient rule: (denominator·(numerator)′ − numerator·(denominator)′)/(denominator)^2; order matters because of the minus sign.

Radians

The angle unit required for the standard clean trig derivatives; using degrees would introduce a conversion factor.

d/dx(sin x)

cos x (when x is in radians).

d/dx(cos x)

−sin x (when x is in radians; the negative sign is essential).

d/dx(e^x)

e^x; the exponential function e^x is its own derivative.

d/dx(ln x)

1/x (for x>0 in the real-number setting).

Velocity

The derivative of position: v(t)=s'(t); measures how fast position changes with respect to time.

Acceleration

The derivative of velocity (or second derivative of position): a(t)=v'(t)=s''(t); measures how fast velocity changes.