Unit 2: Differentiation: Definition and Fundamental Properties

From Average Rate of Change to the Derivative

What “rate of change” really means

A major goal of calculus is to describe how quantities change. Before calculus, you probably worked with average rate of change: how much an output changes per unit change in input over an interval. If a function f(x) represents position (in meters) at time x (in seconds), then the average velocity from x=a to x=b is

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

This same expression is also the slope of the secant line through the points \big(a,f(a)\big) and \big(b,f(b)\big) on the graph. In coordinate form, the same idea is often written as “rise over run”:

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

This works perfectly for a straight (linear) line, but for a curved graph, the slope changes from point to point, so you approximate using a secant line over a small interval.

Why this matters: in real life, many questions are about an instant (speed at exactly 3 seconds, marginal cost at exactly 100 units, slope at a point on a curve). Average rate of change can’t answer “at an instant” by itself because it needs an interval.

The key idea: zooming in to get instantaneous change

To get an instantaneous rate of change at x=a, you shrink the interval. Let the second point be a+h, where h is a small number (positive or negative). The secant slope becomes the difference quotient

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

The closer the two points are, the more accurate the secant slope is as an approximation of the “true” slope at x=a. As h gets closer to 0, the secant line “turns into” the tangent line—provided the graph behaves nicely at that point.

Slopes on curves: secant lines vs tangent lines

For a linear function, slope is constant, so “rise over run” gives the slope everywhere. For a non-linear curve, there isn’t one constant slope, so you approximate the slope at a point by drawing a secant line through two nearby points. The tangent line is the limiting position of these secant lines as the points merge.

Definition of the derivative at a point (instantaneous rate of change)

The derivative of f at x=a is the limit of the difference quotient (if the limit exists):

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

You will also see an equivalent form that approaches a directly:

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

These are the same idea: measure the slope of a secant line and then take the limit as the two points merge. This is the formal version of instantaneous rate of change.

Why this matters: this limit definition is the foundation that justifies all differentiation rules. Even when you later use shortcuts like the power rule, the “truth underneath” is this limit.

Worked example: using the definition to find a derivative value

Find f'(2) for f(x)=x^2.

Start with the definition:

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

Compute each piece:

f(2+h)=(2+h)^2=4+4h+h^2

f(2)=4

Substitute:

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

Simplify the numerator:

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

Factor out h and cancel (this is the critical algebra step):

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

Now take the limit:

f'(2)=4

Interpretation: the tangent line slope to y=x^2 at x=2 is 4.

What can go wrong

A very common error is trying to plug in h=0 too early. In the definition,

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

is undefined at h=0. You must simplify first (usually by factoring and canceling), and only then take the limit.

Exam Focus

• Typical question patterns:
  • “Use the definition of the derivative to find f'(a) for a given function.”
  • “Write the expression for the derivative at x=a as a limit.”
  • “Interpret the derivative as the slope of a tangent line or instantaneous rate of change.”
• Common mistakes:
  • Plugging in h=0 before algebraic simplification.
  • Cancelling incorrectly (you may cancel a factor of h, but never cancel terms across addition).
  • Mixing up the two equivalent forms and substituting the wrong point (for example, using f(x)-f(h) instead of f(a+h)-f(a)).

The Derivative as a Function (Not Just a Number)

From “the slope at one point” to “a new function that gives slopes”

When you compute f'(2), you get one number. But you can repeat the process at every input value and produce a brand-new function, the **derivative function** f'(x), which gives the slope of the tangent line at each x.

Using the limit definition at a general x:

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

Why this matters: thinking of f'(x) as a function lets you analyze how slope changes across an interval, which leads directly to graph analysis and optimization later.

Worked example: deriving f'(x) from first principles

Let f(x)=x^2 again, but now find f'(x).

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

Expand:

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

Simplify:

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

Factor and cancel:

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

Take the limit:

f'(x)=2x

Now you have a rule: at any input x, the tangent slope is 2x.

Notation you must be comfortable with

AP Calculus uses multiple equivalent notations. They look different, but they represent the same derivative concept.

You should also recognize second derivative notation (derivative of the derivative):

A key conceptual point: \frac{dy}{dx} is read “derivative of y with respect to x.” On AP problems, this notation often appears in contexts involving units or rates.

Connecting derivative function to a graph

If f'(x) > 0 on an interval, then f(x) is increasing there. If f'(x) < 0 on an interval, then f(x) is decreasing there. If f'(x) = 0 at a point, the tangent is horizontal (but that alone does not guarantee a max or min).

Exam Focus

• Typical question patterns:
  • “Find f'(x) using the limit definition.”
  • “Given a graph of f, sketch a graph of f' (conceptually).”
  • “Evaluate f'(a) from a table or from a derivative expression.”
• Common mistakes:
  • Treating f'(x) as a single number rather than a function.
  • Confusing f'(a) with f(a).
  • Forgetting that a horizontal tangent (derivative zero) does not automatically mean a local extremum.

Interpreting Derivatives: Slope, Velocity, and Units

The derivative is a meaning-maker

AP problems often test whether you can interpret derivatives, not just compute them. The derivative is best understood as:

• the slope of the tangent line to y=f(x) at a point
• the instantaneous rate of change of f with respect to x

These are the same idea in two languages: geometry (slope) and real-world change (rate).

Units: one of the most tested interpretation skills

If f(x) has units and x has units, then f'(x) has units “units of f per unit of x.”

Example: If s(t) is position in meters and t is seconds, then s'(t) is meters per second.

Motion interpretation (a classic application)

If s(t) is position, then velocity and acceleration are derivatives:

v(t)=s'(t)

a(t)=v'(t)=s''(t)

This reinforces that derivatives can be taken repeatedly, and each derivative has a distinct meaning.

Tangent line approximation (linearization idea, informal)

Near x=a, the graph of f looks almost like its tangent line if f is smooth there. The tangent line at x=a is

L(x)=f(a)+f'(a)(x-a)

You may see questions asking you to approximate values using tangent lines.

Worked example: interpreting a derivative in context

Suppose P(t) is the population of a town (people) at time t (years). If P'(5)=120, then at t=5 years the population is increasing at a rate of 120 people per year.

Common misunderstanding: saying “the population is 120.” That would be P(5)=120, not P'(5)=120.

Worked example: writing a tangent line

Let f(x)=\sqrt{x}. Find the tangent line at x=4.

First compute f(4):

f(4)=2

Compute the derivative (later you’ll justify this with rules):

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

Evaluate at 4:

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

Use point-slope form:

y-2=\frac{1}{4}(x-4)

Simplify if desired:

y=\frac{1}{4}x+1

Interpretation: near x=4, \sqrt{x} is close to \frac{1}{4}x+1.

Exam Focus

• Typical question patterns:
  • “Interpret f'(a) in context, including units.”
  • “Find the equation of the tangent line to f at x=a.”
  • “Use a tangent line to approximate a nearby function value.”
• Common mistakes:
  • Confusing function value with derivative value (quantity vs rate).
  • Dropping or mis-stating units.
  • Using the wrong point in the tangent line formula (using f'(a) as a point, or forgetting \big(a,f(a)\big)).

When Does a Derivative Exist? Differentiability and Continuity

Differentiability is a stronger condition than continuity

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

A function is continuous at x=a if

\lim_{x\to a} f(x)=f(a)

Key relationship:

• If f is differentiable at a, then f is continuous at a.
• The converse is not always true: a function can be continuous but not differentiable.

Geometric situations where differentiability fails

From the tangent-line viewpoint, differentiability can fail when there is no single well-defined tangent slope.

1. Corner: left-hand slope and right-hand slope are finite but different.
2. Cusp: slopes approach infinity but in opposite directions.
3. Vertical tangent: slope becomes infinite (derivative does not exist as a finite number).
4. Discontinuity: if the function jumps, has a hole, or blows up, it cannot be differentiable there.

A crisp way to detect non-differentiability at x=a is to compare one-sided limits of the difference quotient:

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

and

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

If these one-sided limits are not equal (or not finite), then f'(a) does not exist.

Worked example: a corner with an absolute value

Consider f(x)=|x| at x=0.

Compute the difference quotient:

\frac{|h|-|0|}{h}=\frac{|h|}{h}

If h>0, then |h|=h, so the quotient is 1. If h