Differentiation: Definition and Fundamental Properties

What a Derivative Measures

A derivative is calculus’ way of describing how something changes right now. In algebra, you describe patterns with formulas. In calculus, you go one level deeper and ask how sensitive the output is to small changes in the input.

There are two big, equivalent interpretations you should build from the start:

1. Slope interpretation (geometric): the derivative is the slope of the tangent line to a graph at a point.
2. Rate interpretation (applied): the derivative is the instantaneous rate of change of one quantity with respect to another.

AP problems move back and forth between these interpretations constantly.

Rates of change: average vs instantaneous

There are two main ways to talk about “rate of change.”

Over an interval from x=a to x=b, the average rate of change is

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

In coordinate form, this is the familiar slope formula

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

In many contexts, this is “rate of change over an interval of time.”

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Geometrically, \frac{f(b)-f(a)}{b-a} is the slope of the **secant line** through the two points \left(a,f(a)\right) and \left(b,f(b)\right).

To capture change at a specific point in time, you use the instantaneous rate of change, which comes from the same difference quotient but with a limit as h \to 0.

Slopes on curves: secant lines and tangent lines

For a linear function, slope is “rise over run” and is constant. For a curved graph, the slope changes from point to point, so you approximate the slope near a point using a secant line.

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The closer the two points are, the more accurate the secant slope becomes as an estimate of the “true” slope at that point.

A tangent line is the limiting position of these secant lines as the second point moves in toward the first. It is commonly described as a line that “touches the curve at exactly one point,” but the most reliable definition is this: it is the line whose slope equals the limiting slope of nearby secants (and a function can cross its tangent line).

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From average change to instantaneous change (difference quotient)

Pick a nearby point a+h and compute the slope from a to a+h:

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

As h gets closer to 0, the second point slides toward the first point. If the slopes approach a single value, that limiting slope is the derivative.

Real-world meaning (why you should care)

Derivatives show up wherever a small input change produces an output change:

• Motion: if s(t) is position, then s'(t) is instantaneous velocity.
• Economics: if C(x) is cost to produce x items, then C'(x) is marginal cost (approximate cost of producing one more item).
• Population: if P(t) is population, then P'(t) is the instantaneous growth rate.

The key idea is that a derivative is a local measurement. It tells you what the function is doing near a point.

Example: average rate of change as a secant slope

Let f(x)=x^2. Compute the average rate of change from x=1 to x=3.

\frac{f(3)-f(1)}{3-1}=\frac{9-1}{2}=4

That means over that interval, f(x) increases about 4 units in output per 1 unit in input.

Example: zooming in toward an instantaneous rate

Using the same f(x)=x^2, look at slopes from x=2 to x=2+h:

\frac{f(2+h)-f(2)}{h}=\frac{(2+h)^2-4}{h}=\frac{4+4h+h^2-4}{h}=\frac{4h+h^2}{h}=4+h

If h is very small, the slope is very close to 4. The limiting value as h \to 0 will be the derivative at x=2.

Exam Focus

• Typical question patterns
  • Given a table or context, compute an average rate of change and interpret units.
  • Given a graph, estimate slopes of secant lines and reason what happens as points get closer.
  • Translate wording like “instantaneous rate at x=a” into the limit idea.
• Common mistakes
  • Mixing up \frac{f(b)-f(a)}{b-a} (average on an interval) with the derivative (instantaneous at a point).
  • Forgetting that the denominator in \frac{f(a+h)-f(a)}{h} is the change in input (so it must match the function’s input variable and units).
  • Treating “tangent line” as a line that merely touches once; many functions cross their tangent line.

The Limit Definition of the Derivative

The derivative is defined using a limit because we need a precise way to capture what happens as the interval shrinks to zero. The limit ensures we’re not just picking a tiny interval; we’re describing what value the slopes approach.

Definition at a point

The derivative of f at x=a, written f'(a), is

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

This expression is called a difference quotient. It is the slope of a secant line, taken to the limit where the secant becomes a tangent.

There’s an equivalent form you’ll also see:

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

These represent the same idea with different approach variables.

Derivative as a function

If you replace the specific number a with a variable x, you get the derivative function:

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

This matters because problems often ask for an expression for f'(x) (a new function), not just a single slope at one point.

When the derivative does not exist

The limit must exist as a real number. If the left-hand and right-hand approaches disagree, or if the slope becomes infinite, then f'(a) does not exist.

Common reasons a derivative fails:

• discontinuity (a jump, hole, or asymptote)
• corner or cusp (left and right slopes don’t match or blow up differently)
• vertical tangent (slope approaches \pm\infty)

Worked example: derivative from the definition

Find f'(x) for f(x)=x^2 using the limit definition.

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

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

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

Worked example: derivative at a point from the definition (conjugates)

Let f(x)=\sqrt{x}. Find f'(4) from first principles.

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

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

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

Interpreting: near x=4, \sqrt{x} increases at about 0.25 output units per 1 input unit.

Exam Focus

• Typical question patterns
  • “Use the definition of the derivative to find f'(x)” for a simple function.
  • Compute f'(a) exactly from the limit definition (often with algebraic simplification like conjugates).
  • Decide whether a derivative exists at a point based on one-sided limits.
• Common mistakes
  • Plugging in h=0 too early (you must simplify before evaluating the limit).
  • Losing parentheses when expanding f(x+h) (a very common source of sign errors).
  • Forgetting to use the conjugate when radicals cause an indeterminate form.

Derivative Notation and the Meaning of \frac{dy}{dx}

Calculus uses multiple notations for derivatives because they emphasize different interpretations. You should be able to read all of them fluently.

Notation you will see (and what it means)

If y=f(x), then the derivative can be written in several equivalent ways:

You will also see second derivative notation, which measures how the first derivative changes:

What \frac{dy}{dx} is (conceptually)

The notation \frac{dy}{dx} comes from the idea “change in output over change in input,” because derivatives start from the difference quotient and then become instantaneous via a limit. In a strict algebraic sense, \frac{dy}{dx} is not simply a fraction you can always treat like ordinary division, but it often behaves like one in later topics.

For Unit 2, the safest interpretation is:

• \frac{dy}{dx} represents the slope of the tangent line to the graph of y versus x.
• Units of \frac{dy}{dx} are “units of y per unit of x.”

Derivative values vs derivative function

It’s important not to blur these two:

• f'(a) is a single number (slope at x=a).
• f'(x) is a new function giving the slope at every x where it exists.

Example: interpreting units

If s(t) measures distance in meters and t is time in seconds, then s'(t) has units meters per second.

If C(x) measures cost in dollars and x is number of items, then C'(x) has units dollars per item.

Example: tangent slope from notation

Suppose you’re told f'(3)=-2. That means the tangent line slope at x=3 is -2, and near x=3, increasing x by about 1 decreases f(x) by about 2.

Exam Focus

• Typical question patterns
  • Given f'(a), interpret what it says about the function’s behavior and include units.
  • Convert between notations: recognize that \frac{d}{dx}[f(x)] means “differentiate f(x).”
  • Use derivative information to write a tangent line equation (often paired with a function value).
• Common mistakes
  • Treating f'(x) and f(x) as interchangeable (they are different functions).
  • Dropping units in context problems (units are part of the meaning of a derivative).
  • Confusing f'(a) with f(a), especially when both are provided in a word problem.

Differentiability and Continuity

The derivative is defined by a limit, so differentiability is closely linked to continuity, but they are not the same idea.

What it means to be differentiable

A function f is differentiable at x=a if f'(a) exists as a finite real number. It is differentiable on an interval if it’s differentiable at every point in that interval.

Geometrically, differentiable at a point means the graph has a well-defined (finite) tangent slope there.

Differentiability implies continuity

A fundamental fact is:

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

But the converse is not guaranteed:

• A function can be continuous at a but not differentiable at a.

Where differentiability fails (and what it looks like)

Even if a function is continuous, the slope might fail to settle to one value.

1. Corner: left-hand slope and right-hand slope are finite but unequal (for example, |x| at 0).
2. Cusp: the slope becomes infinite in different ways from left and right.
3. Vertical tangent: slopes approach \pm\infty from both sides in the same way.
4. Discontinuity: if the function is not continuous, it cannot be differentiable.

A precise way to test “left vs right” uses one-sided derivatives:

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

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

If both exist and are equal, then f'(a) exists.

Example: a continuous function that is not differentiable

Let f(x)=|x|. Compute slopes from the definition at x=0.

For h>0:

\frac{f(0+h)-f(0)}{h}=\frac{|h|-0}{h}=\frac{h}{h}=1

For h