Unit 2: Differentiation: Definition and Fundamental Properties

Interpreting the Derivative: Rates of Change, Secant Slopes, and Tangent Slopes

When you first meet the derivative, it helps to think of it as answering one powerful question: how fast is something changing right now? In Algebra, you often focused on an average rate of change across an interval, but Calculus builds a precise way to talk about an instantaneous rate of change at a single input value.

Average rate of change (difference quotient)

For a function f(x), the average rate of change from x=a to x=b is the slope of the secant line through \big(a,f(a)\big) and \big(b,f(b)\big):

\frac{f(b)-f(a)}{b-a}

You may also see this described informally as:

\frac{y_2-y_1}{x_2-x_1}

This is “change in output per change in input” over an interval, not at a single instant.

Instantaneous rate of change (the derivative)

The other way of describing rate of change is at a specific point (like a speedometer reading at one moment). For a curved graph, you can’t just do “rise over run” using two points on the curve to get the slope at exactly one point. Instead, you approximate with secant lines and then take a limit.

From secant lines to a tangent line

Pick a point at x=a and a nearby point at x=a+h. The slope of the secant line is

\frac{f(a+h)-f(a)}{(a+h)-a}

which simplifies to

\frac{f(a+h)-f(a)}{h}

As the two points get closer together (as h approaches 0), the secant slope becomes more accurate, and in the limit it becomes the slope of the tangent line—when that tangent line exists.

A common informal description is that the tangent line “touches” the curve at one point, but the deeper idea is local agreement in direction (slope). A tangent line can intersect the curve again elsewhere; what matters is that it matches the curve’s slope at the point of tangency.

Why this matters (applications)

This interpretation is why derivatives show up everywhere:

• In physics, the derivative becomes velocity and acceleration.
• In economics, it models marginal cost and marginal revenue.
• In biology, it can represent population growth rates.
• In geometry, it gives the slope of a curve at a point.

Example (conceptual): average vs instantaneous

Suppose s(t) measures position (meters) of a moving object at time t (seconds).

Average velocity from t=2 to t=5 is

\frac{s(5)-s(2)}{5-2}

Instantaneous velocity at t=2 (if it exists) is the derivative s'(2), defined via a limit of average velocities over shrinking intervals.

Exam Focus

• Typical question patterns:
  • Interpret f'(a) as slope of the tangent line or instantaneous rate of change in context (with units).
  • Compare average rate of change on an interval to instantaneous rate at an endpoint.
  • Relate “increasing/decreasing” language to the sign of the derivative.
• Common mistakes:
  • Treating average rate of change and derivative as the same thing.
  • Forgetting units: if f is in dollars and x is in years, then f' is dollars per year.
  • Thinking the tangent line must “touch only once” (tangent means matching slope locally, not “touching once”).

The Limit Definition of the Derivative

The derivative is defined using a limit because “instantaneous” change cannot be captured by a finite difference. You approximate the rate of change with secant slopes, then take the limit as the interval shrinks to zero.

Two equivalent limit definitions

At a point x=a, the derivative of f at a can be written as

f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}

An equivalent form is

f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}

Both express the same idea: a nearby point approaches a.

What it means for the derivative to exist

The derivative f'(a) exists if the defining limit exists and is finite. Intuitively, the derivative exists if, when you zoom in near x=a, the graph looks more and more like a straight line. Differentiability fails at corners, cusps, vertical tangents (infinite slope), and at any discontinuity.

One-sided derivatives

Sometimes you approach from one side:

Right-hand derivative at a:

\lim_{h\to 0^+}\frac{f(a+h)-f(a)}{h}

Left-hand derivative at a:

\lim_{h\to 0^-}\frac{f(a+h)-f(a)}{h}

For f'(a) to exist, both one-sided derivatives must exist and be equal.

Worked example 1: derivative from the definition

Find the derivative of f(x)=x^2 using the limit definition.

Start with

f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

Compute

f(x+h)=(x+h)^2

Substitute:

f'(x)=\lim_{h\to 0}\frac{(x+h)^2-x^2}{h}

Expand the numerator:

(x+h)^2-x^2=x^2+2xh+h^2-x^2=2xh+h^2

So

f'(x)=\lim_{h\to 0}\frac{2xh+h^2}{h}

Factor out and cancel h:

\frac{2xh+h^2}{h}=\frac{h(2x+h)}{h}=2x+h

Now take the limit:

f'(x)=\lim_{h\to 0}(2x+h)=2x

Worked example 2: a derivative at a specific point

Let f(x)=\sqrt{x}. Find f'(4) from the limit definition.

f'(4)=\lim_{h\to 0}\frac{\sqrt{4+h}-\sqrt{4}}{h}

f'(4)=\lim_{h\to 0}\frac{\sqrt{4+h}-2}{h}

Multiply by the conjugate:

f'(4)=\lim_{h\to 0}\frac{\sqrt{4+h}-2}{h}\cdot\frac{\sqrt{4+h}+2}{\sqrt{4+h}+2}

Then

f'(4)=\lim_{h\to 0}\frac{(4+h)-4}{h(\sqrt{4+h}+2)}

Simplify:

f'(4)=\lim_{h\to 0}\frac{h}{h(\sqrt{4+h}+2)}

Cancel h (valid for h\neq 0, which is fine inside a limit process):

f'(4)=\lim_{h\to 0}\frac{1}{\sqrt{4+h}+2}

Substitute h=0:

f'(4)=\frac{1}{\sqrt{4}+2}=\frac{1}{4}

Exam Focus

• Typical question patterns:
  • Compute f'(a) using the limit definition (often requiring algebra like factoring or conjugates).
  • Identify which limit expression matches the derivative at a point.
  • Use left and right definitions to test differentiability at a “sharp” point.
• Common mistakes:
  • Plugging in h=0 before simplifying, then stopping at 0/0.
  • Algebra slips: expanding f(x+h) incorrectly.
  • Canceling terms incorrectly (you can cancel a factor of h, not a term that is added).

Derivative Notation and What the Symbols Mean

Calculus uses several notations because different contexts emphasize different ideas: sometimes you want to highlight a function name, sometimes a dependent variable, and sometimes the “operator” viewpoint.

Notation reference

A key conceptual point is that f'(x) is a new function (it takes inputs and produces slopes/rates), while f'(a) is a single number (the slope/rate at one specific input).

Average rate vs derivative language

Average rate of change on [a,b]:

\frac{f(b)-f(a)}{b-a}

Instantaneous rate of change at a:

f'(a)

On AP questions, the wording often shifts between “rate of change,” “slope,” “velocity,” “marginal,” or “derivative.” These are often pointing to the same underlying idea.

Interpreting the units

Units are one of the best ways to check your reasoning.

• If x is hours and f(x) is miles, then f'(x) is miles per hour.
• If t is seconds and s(t) is meters, then s'(t) is meters per second.

Example: interpreting notation in context

Let C(q) be the cost in dollars to produce q items.

• C'(50) is the instantaneous rate at which cost changes when producing 50 items, with units dollars per item.
• If C'(50)=3.2, then near 50 items, producing one additional item increases cost by about 3.20.

Exam Focus

• Typical question patterns:
  • Translate between f'(a), \frac{dy}{dx}, and \frac{d}{dx}f(x).
  • Interpret f'(a) in a word problem (including units).
  • Distinguish between f'(x) (function) and f'(a) (value).
• Common mistakes:
  • Writing f'(x) when the question asks for f'(3) (leaving the answer as a function instead of a number).
  • Dropping units or mixing them up.
  • Confusing \frac{dy}{dx} with \frac{\Delta y}{\Delta x} (instantaneous vs average).

Estimating Derivatives from Graphs and Tables

Many AP problems give you data rather than a formula. In that setting, you estimate the derivative by estimating a tangent slope (from a graph) or a limiting secant slope (from a table).

Estimating from a graph

If you’re asked to estimate f'(a) from the graph of f, you are being asked for the slope of the tangent line at x=a.

A reliable strategy is:

1. Locate the point \big(a,f(a)\big).
2. Draw the tangent line that matches the curve’s local direction at that point.
3. Pick two convenient points on your drawn tangent line (they do not need to be points on the curve) and compute slope.

Slope is always

\frac{\text{rise}}{\text{run}}

Because this is an estimate, your answer is often written as “approximately” and may be a decimal.

Estimating from a table

If you have values at a and a+h, a one-sided secant estimate is

\frac{f(a+h)-f(a)}{h}

If you have values on both sides, a common and often more accurate estimate is the symmetric difference quotient:

f'(a)\approx \frac{f(a+h)-f(a-h)}{2h}

Deciding whether the derivative is positive, negative, or zero

Even without an exact slope value, you can often determine the sign.

• If f is increasing at a, then f'(a) is positive.
• If f is decreasing at a, then f'(a) is negative.
• If the tangent is horizontal, then f'(a)=0.

A horizontal tangent gives a critical point candidate, not automatically a maximum or minimum.

Worked example: table-based estimate

A table gives values of f near x=2:

Estimate f'(2) using a symmetric difference quotient with h=0.1:

f'(2)\approx \frac{f(2.1)-f(1.9)}{2(0.1)}

Substitute values:

f'(2)\approx \frac{6.08-5.34}{0.2}

Compute:

f'(2)\approx \frac{0.74}{0.2}=3.7

Interpretation: near x=2, the function increases by about 3.7 output units per 1 input unit.

Worked example: graph-based sign reasoning

If a graph shows f decreasing steeply at x=1, then f'(1) is negative with relatively large magnitude. If the curve is almost flat there, f'(1) is negative but closer to zero.

Exam Focus

• Typical question patterns:
  • Estimate f'(a) from a table using secant slopes (often symmetric).
  • Estimate f'(a) from a graph by drawing a tangent and computing its slope.
  • Determine where f'(x) is positive/negative/zero from the shape of f.
• Common mistakes:
  • Using points on the curve instead of points on the tangent line when estimating from a graph.
  • Mixing up f(a) with f'(a) (reporting the y-value instead of the slope).
  • Forgetting that the slope is “rise over run” and accidentally inverting it.

Differentiability and Continuity

It’s tempting to assume that if a graph is continuous (no breaks), then it must have a derivative. Calculus separates these ideas.

• Continuity is about whether the graph has holes/jumps at a point.
• Differentiability is about whether the graph has a well-defined (finite) tangent slope at a point.

Key relationship

If f is differentiable at x=a, then f is continuous at x=a.

The converse is not always true: a function can be continuous but not differentiable.

Where differentiability fails (even if continuous)

A function may be continuous at a but not differentiable there if it has:

1. A corner (left-hand and right-hand slopes are finite but different).
2. A cusp (slopes become infinite in opposite directions).
3. A vertical tangent (slope becomes infinite).

Differentiability also fails at discontinuities (jumps, holes, vertical asymptotes), since differentiability implies continuity.

How to reason using one-sided derivatives (especially for piecewise functions)

A useful workflow at a point where a formula changes is:

1. Check continuity at the point.
2. Compute the left-hand derivative and right-hand derivative.
3. If both exist and are equal, the derivative exists.

Skipping step 1 can waste time: if the function is discontinuous, it is automatically not differentiable.

Worked example: corner in an absolute value function

Consider f(x)=|x| at x=0. The graph has a sharp corner at the origin.

For x>0, |x|=x so the slope is 1.

For x