Unit 2: Differentiation: Definition and Fundamental Properties

Average rate of change

The change in function output over a nonzero interval, computed as a slope between two points (a secant slope).

Difference quotient

The expression (\frac{f(a+h)-f(a)}{h}) (with (h\neq 0)), used to compute average rates of change and define the derivative via a limit.

Secant line

A line that intersects a curve at two points; its slope equals the average rate of change over the interval between those points.

Secant slope

The slope of the secant line through ((a,f(a))) and ((a+h,f(a+h))), given by (\frac{f(a+h)-f(a)}{h}).

Instantaneous rate of change

The rate of change at a single input value, defined as the limit of average rates of change as the interval shrinks to zero.

Tangent line

A line that touches a curve at a point and locally matches the curve’s direction; its slope is the derivative at that point (when it exists).

Slope of the tangent line

The limit of secant slopes as the second point approaches the first; equals (f'(a)) at (x=a) if the derivative exists.

Derivative at a point

(f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}), the instantaneous rate of change (slope) at (x=a).

Differentiable at (a)

A function is differentiable at (a) if (\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}) exists and is finite.

Equivalent derivative limit form

(f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}), expressing the derivative using (x\to a) instead of (h\to 0).

Indeterminate form (0/0)

A limit form that occurs when substituting directly gives (\frac{0}{0}); it signals you must simplify before evaluating the limit.

Derivative function

The function (f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}), giving the tangent slope for each input where it exists.

Prime notation

Derivative notation like (f'(x)) or (y') indicating the derivative with respect to the input variable.

Leibniz notation

Notation (\frac{dy}{dx}) emphasizing the derivative as a rate of change (with units).

Derivative operator notation

Notation (\frac{d}{dx}[f(x)]) that clearly indicates which function is being differentiated.

Second derivative

The derivative of the derivative, written (f''(x)) or (y'').

Point-slope form (tangent line)

Equation form (y-f(a)=f'(a)(x-a)) for the tangent line to (y=f(x)) at (x=a).

Units of a derivative

If (f) has units and (x) has units, then (f') has units (\frac{\text{units of }f}{\text{units of }x}) (e.g., meters per second).

Velocity as a derivative

If (s(t)) is position, then instantaneous velocity is (v(t)=s'(t)), the rate of change of position with respect to time.

Symmetric difference quotient

An estimate for (f'(a)) using nearby values: (f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}), often more accurate than one-sided estimates.

Left-hand derivative

(f'-(a)=\lim{h\to 0^-}\frac{f(a+h)-f(a)}{h}), the derivative approached from inputs less than (a).

Right-hand derivative

(f'+(a)=\lim{h\to 0^+}\frac{f(a+h)-f(a)}{h}), the derivative approached from inputs greater than (a).

Derivative exists criterion

(f'(a)) exists only if both one-sided derivatives exist and are equal.

Differentiability implies continuity

The theorem that if (f) is differentiable at (a), then (f) must be continuous at (a).

Continuous but not differentiable

A situation where a function has no break at a point but still lacks a derivative there (commonly due to a corner/cusp or vertical tangent).

Corner

A point where the left-hand and right-hand tangent slopes approach different finite values, so the derivative does not exist.

Cusp

A sharp point where slopes become infinite in opposite ways (or otherwise fail to match), so the derivative does not exist.

Vertical tangent

A point where the slope becomes unbounded (infinite); in AP Calculus language, the derivative does not exist there.

Discontinuity

A hole, jump, or asymptote in a graph; a function cannot be differentiable at a point where it is discontinuous.

Increasing function and derivative sign

Where (f) is increasing, tangent slopes are positive, so (f'(x)>0).

Decreasing function and derivative sign

Where (f) is decreasing, tangent slopes are negative, so (f'(x)<0).

Horizontal tangent

A tangent line with slope 0 at (x=a); equivalently (f'(a)=0).

Constant rule

If (f(x)=k) (constant), then (\frac{d}{dx}[k]=0).

Constant multiple rule

(\frac{d}{dx}[c f(x)] = c f'(x)), allowing constants to factor out of derivatives.

Sum rule

(\frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x)).

Difference rule

(\frac{d}{dx}[f(x)-g(x)]=f'(x)-g'(x)).

Linearity of differentiation

The collection of rules (constant multiple, sum, difference) showing derivatives distribute over addition/subtraction and scale by constants.

Power rule

For (f(x)=x^n), (\frac{d}{dx}[x^n]=n x^{n-1}) (multiply by the exponent and reduce the power by 1).

Negative exponent differentiation

Applying the power rule to (x^{-n}) gives (-n x^{-(n+1)}), often rewritten with positive exponents as a rational expression.

Product rule

If (f(x)=u\,v), then (f'(x)=u\,v' + v\,u').

Quotient rule

If (f(x)=\frac{u}{v}), then (f'(x)=\frac{v u' - u v'}{v^2}).

Quotient rule numerator order

In (\frac{v u' - u v'}{v^2}), the subtraction order matters; reversing it changes the sign of the derivative.

Trigonometric derivative (sine)

(\frac{d}{dx}[\sin x]=\cos x) (valid when (x) is measured in radians).

Trigonometric derivative (cosine)

(\frac{d}{dx}[\cos x]=-\sin x) (valid when (x) is measured in radians).

Radians requirement for trig derivatives

The standard trig derivative formulas assume angles are in radians; using degrees breaks these formulas without conversion.

Exponential memory derivative

(\frac{d}{dx}[e^x]=e^x).

Logarithmic memory derivative

(\frac{d}{dx}[\ln x]=\frac{1}{x}) (real-valued domain typically (x>0)).

Conjugate method (limits)

An algebra technique (multiply by a conjugate) used to simplify difference quotients involving radicals to remove (0/0).

Factor-and-cancel method (limits)

An algebra technique used to simplify difference quotients (especially polynomials) by factoring out a common factor (often (h)) and canceling it.

Canceling factors vs. canceling terms

You may cancel only common factors (e.g., (\frac{h(h+1)}{h})), not individual terms in a sum (e.g., you cannot cancel in (\frac{h^2+1}{h})).