Unit 2 (AP Calculus AB): Derivatives from Limits, Meaning, Estimation, Differentiability, and Core Rules

Average rate of change

The slope over an interval x=a to x=b: (f(b)-f(a))/(b-a); represents one overall rate for the whole interval.

Secant line

A line through two points on a curve, such as (a,f(a)) and (b,f(b)); its slope gives the average rate of change.

Difference quotient

An expression for average rate of change, especially from a to a+h: (f(a+h)-f(a))/h.

Instantaneous rate of change

How fast f(x) is changing at a specific input; computed as a derivative value f'(a) (not an average over an interval).

Limit

A value that an expression approaches as the input approaches a point (e.g., h\to 0), used to define derivatives without direct substitution that would cause division by zero.

Derivative (at a point)

The instantaneous rate of change of f at x=a, equal to the slope of the tangent line there, denoted f'(a).

Limit definition of the derivative

f'(a)=\lim_{h\to 0} \frac{f(a+h)-f(a)}{h}; the core (first-principles) definition of a derivative.

Tangent line

A line that touches the curve at (a,f(a)) and gives the best local linear approximation; its slope is f'(a) when the derivative exists.

Point-slope form

A line form used to write tangent lines: y-y1=m(x-x1) (e.g., y-f(a)=f'(a)(x-a)).

Tangent line (local model) L(x)

The linear function that approximates f near a: L(x)=f(a)+f'(a)(x-a).

Local linear approximation

Using the tangent line at x=a to approximate f(x) for x near a (works best for small changes near a).

Differential approximation

For small \Delta x near a, the output change is approximately \Delta y\approx f'(a)\Delta x.

Units of a derivative

If f(x) has output units and x has input units, then f'(x) has units (output units)/(input units) (a built-in error check).

Velocity (as a derivative)

If s(t) is position vs. time, then s'(t) is velocity: an instantaneous rate of change with units like meters per second.

Marginal cost

If C(x) is cost vs. number of items, then C'(x) is marginal cost: dollars per item at that production level.

Derivative notation dy/dx

Leibniz-style notation emphasizing “rate of change of y with respect to x,” often used in applied interpretation.

Prime notation

A compact derivative notation: f'(x) is the derivative function and f'(a) is the derivative value at x=a.

Leibniz notation df/dx

Another common way to write f'(x); highlights the variable of differentiation and is widely used in calculus contexts.

Second derivative

The derivative of the derivative, written f''(x) or d^2y/dx^2; represents the rate of change of f'(x).

Higher-order derivatives notation f^{(n)}(x)

The nth derivative of f (e.g., third derivative f'''(x)); found by differentiating repeatedly.

One-sided derivatives

Derivatives defined using only one side: f'-(a) from the left and f'+(a) from the right; f'(a) exists only if they are equal.

Differentiable at a point

A function is differentiable at x=a if the limit in the derivative definition exists as a real number (left and right derivative agree).

Differentiable on an interval

A function is differentiable on an interval if it is differentiable at every interior point of that interval.

Continuity

A function is continuous at x=a if the limit of f(x) as x\to a equals f(a) (no break at a).

Differentiability implies continuity theorem

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

Corner

A point where the left and right slopes are finite but different; the function may be continuous but is not differentiable there (e.g., |x| at 0).

Cusp

A very sharp point where slopes become infinite with opposite signs; the derivative does not exist there as a finite real number.

Vertical tangent

A point where the slope becomes infinite; the derivative is not a finite real number, so f is not differentiable there.

Discontinuity

A break, jump, or hole in the graph at x=a; if f is not continuous at a, it cannot be differentiable at a.

Piecewise differentiability

For a piecewise function to be differentiable at a junction, it must be continuous there and have matching left and right derivative values (smooth connection).

Continuity condition at a junction

For a piecewise function at x=a, the pieces must meet: the left-hand limit and right-hand limit equal f(a).

Slope condition at a junction

After continuity is ensured, the left-hand derivative and right-hand derivative at the junction must be equal for differentiability.

Linearity of the derivative

Differentiation distributes over addition/subtraction and allows constants to factor out: derivatives of sums/differences are sums/differences of derivatives.

Constant multiple rule

\frac{d}{dx}(c f(x))=c f'(x); you can “pull out” a constant when differentiating.

Sum rule

\frac{d}{dx}(f(x)+g(x))=f'(x)+g'(x); derivative of a sum is the sum of derivatives.

Constant rule

\frac{d}{dx}(c)=0; a constant function has zero rate of change everywhere.

Power rule

\frac{d}{dx}(x^n)=n x^{n-1} (where defined); “multiply down and decrease the power.”

Polynomial term-by-term differentiation

To differentiate a polynomial, apply the power rule to each term separately; constant terms disappear.

Product rule

For differentiable u,v: \frac{d}{dx}(uv)=u'v+uv' (not u'v').

Quotient rule

For differentiable u,v with v\ne 0: \frac{d}{dx}(u/v)=\frac{u'v-uv'}{v^2}.

Simplify before differentiating (cancellation caution)

Algebraic simplification can reduce work, but canceling factors may hide holes/domain restrictions; simplified and original forms match only where the original is defined.

Symmetric difference quotient

A table-based estimate of f'(a) using points on both sides: \frac{f(a+h)-f(a-h)}{2h}; often more accurate than one-sided estimates.

Forward difference quotient

A table-based estimate using values to the right of a: f'(a)\approx \frac{f(a+h)-f(a)}{h}.

Backward difference quotient

A table-based estimate using values to the left of a: f'(a)\approx \frac{f(a)-f(a-h)}{h}.

Derivative of \sin x

\frac{d}{dx}(\sin x)=\cos x.

Derivative of \cos x

\frac{d}{dx}(\cos x)=-\sin x (watch the negative sign).

Derivative of \tan x

\frac{d}{dx}(\tan x)=\sec^2 x (can be derived using tan x=\sin x/\cos x and the quotient rule).

Derivative of e^x

\frac{d}{dx}(e^x)=e^x; the natural exponential’s rate of change equals its value.

Derivative of a^x

For a>0 and a\ne 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}.