Unit 2: Differentiation: Definition and Fundamental Properties

Average rate of change

The change in a function’s output divided by the change in input over an interval; equal to the slope of a secant line.

Secant line

A line that passes through two points on a curve, typically at x=a and x=a+h.

Secant slope

The slope of the secant line through (a,f(a)) and (a+h,f(a+h)); represents average rate of change on that interval.

Difference quotient

The expression (f(a+h)-f(a))/h; the standard form of the average rate of change from a to a+h.

Two-point slope formula

The slope between two points (x1,y1) and (x2,y2): (y2−y1)/(x2−x1).

Instantaneous rate of change

The rate at which a quantity changes at an exact input value; computed using the derivative (a limit of secant slopes).

Tangent line

The line that matches a curve’s local direction at a point (it may cross the curve; “touches once” is not the real definition).

Tangent slope

The slope of the tangent line at a point; equals the derivative value f'(a) when it exists.

Derivative

A mathematical quantity giving instantaneous rate of change (and tangent slope) defined precisely using a limit.

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 limit form (x→a form)

f'(a)=lim_{x→a} (f(x)−f(a))/(x−a); equivalent to the h→0 form via x=a+h.

Derivative at a point

f'(a), a single number representing the slope/rate at x=a (not a function value).

Derivative function

f'(x), a new function that outputs the slope (instantaneous rate) at each x.

Differentiable at a point

A function is differentiable at x=a if the limit defining f'(a) exists and is finite.

One-sided derivatives

Derivatives computed using h→0^+ (right) and h→0^- (left); both must match for f'(a) to exist.

Right-hand derivative

lim_{h→0^+} (f(a+h)−f(a))/h; uses points to the right of a.

Left-hand derivative

lim_{h→0^-} (f(a+h)−f(a))/h; uses points to the left of a.

Differentiability implies continuity

If f is differentiable at a, then f must be continuous at a (no hole, jump, or asymptote at that point).

Continuity does not imply differentiability

A function can be continuous at a point but still not have a derivative there (e.g., a corner or cusp).

Corner

A point where the left-hand and right-hand slopes approach different finite values, so f'(a) does not exist.

Cusp

A sharp point where slopes approach infinity with opposite signs, so the derivative does not exist.

Vertical tangent

A point where the tangent line is vertical and the slope is infinite/undefined, so the derivative does not exist.

Discontinuity

A break in a function (hole, jump, asymptote); prevents differentiability at that point.

Horizontal tangent

A tangent line with slope 0; indicates f'(a)=0 at that point (often at smooth local maxima/minima).

Increasing (via f'(x)>0)

If f'(x)>0 on an interval, the function is increasing there (at least locally).

Decreasing (via f'(x)<0)

If f'(x)<0 on an interval, the function is decreasing there (at least locally).

Units of the derivative

If f has units and x has units, then f'(x) has units (units of f)/(units of x), e.g., meters per second.

Slope interpretation of derivative

For y=f(x), f'(a) is the slope of the tangent line to the graph at x=a.

Rate-of-change interpretation of derivative

If f models a changing quantity, f'(a) is the instantaneous rate of change of that quantity at x=a.

Local linearity

The idea that a differentiable function behaves approximately like a line near a point x=a.

Linearization (tangent line approximation)

Approximating f near x=a by L(x)=f(a)+f'(a)(x−a).

Tangent line formula (using derivative)

The tangent line at x=a: y=f(a)+f'(a)(x−a), using the point (a,f(a)) and slope f'(a).

Normal line

A line perpendicular to the tangent line; if tangent slope is f'(a)≠0, normal slope is −1/f'(a).

Point-slope form

Line form y−y1=m(x−x1), commonly used to write tangent lines using (a,f(a)) and m=f'(a).

Constant rule

If f(x)=k (constant), then f'(x)=0.

Constant multiple rule

d/dx[ c·f(x) ] = c·f'(x) for a constant c.

Sum rule

d/dx[ f(x)+g(x) ] = f'(x)+g'(x).

Difference rule

d/dx[ f(x)−g(x) ] = f'(x)−g'(x).

Derivative of x (identity function)

If f(x)=x, then f'(x)=1.

Power rule

For positive integers n, d/dx[x^n]=n·x^{n−1} (“multiply down and decrease the power”).

Polynomial differentiation

Differentiate a polynomial term-by-term using linearity and the power rule.

Product rule

If f=uv, then f' = u·v' + v·u' (mnemonic: “1d2 + 2d1”).

Quotient rule

If f=u/v, then f' = (v·u' − u·v')/v^2 (“low d high minus high d low over low squared”).

Leibniz notation (dy/dx)

A common notation for the derivative emphasizing “change in y per change in x” (treat as a single derivative symbol).

Derivative evaluation notation (at a point)

The notation (dy/dx)|_{x=a} means the derivative evaluated at x=a.

Second derivative

The derivative of the derivative, written f''(x) or y''; represents how the rate of change itself changes.

Derivative of sin(x)

d/dx[sin(x)] = cos(x) (with x measured in radians).

Derivative of cos(x)

d/dx[cos(x)] = −sin(x) (negative sign is essential; x in radians).

Derivative of e^x

d/dx[e^x] = e^x.

Derivative of ln(x)

d/dx[ln(x)] = 1/x (in real numbers, requires x>0).