Unit 2: Differentiation: Definition and Fundamental Properties

Secant Slopes, Tangent Slopes, and Rates of Change

Before you ever write a limit, it helps to be clear about what you are trying to measure. A huge portion of calculus is about describing how a quantity changes.

• If you know how far a car traveled over 2 hours, you can compute an average speed.
• But if you want the speed at exactly 3:17 PM, you need an instantaneous speed.

That “instantaneous” idea is the motivation for the derivative.

Average rate of change (secant slope)

Suppose you have a function f and pick two inputs x=a and x=a+h where h is nonzero. The points on the graph are \left(a,f(a)\right) and \left(a+h,f(a+h)\right). The slope of the line through these two points is the **secant slope**, and it equals the **average rate of change** of f over that interval:

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

which simplifies to the famous difference quotient:

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

In “two-point” language, average rate of change is also the familiar slope formula:

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

Instantaneous rate of change (tangent slope)

For a straight (linear) function, slope is “rise over run” and is the same everywhere. For a curved graph, slope changes from point to point, so we approximate the slope near a point using a secant line, and then refine that idea.

A tangent line is the line that matches the curve’s direction at a point. You’ll sometimes hear it described as “touching the curve at exactly one point,” but the important idea is matching the curve’s local direction. In fact, some curves can cross their tangent lines.

The key move is:

• start with a secant slope using a and a+h
• let the second point approach the first point (so h \to 0)

That “let it approach” step is exactly what limits are for.

Worked example: average rate of change vs. “instantaneous-ish”

Let f(x)=x^2.

1) Average rate of change from x=2 to x=5:

\frac{f(5)-f(2)}{5-2}=\frac{25-4}{3}=7

2) Average rate of change from x=2 to x=2.1:

\frac{f(2.1)-f(2)}{2.1-2}=\frac{4.41-4}{0.1}=4.1

Over a big interval the average slope was 7, but very close to 2 it’s around 4.1, suggesting the instantaneous slope at x=2 is near 4.

What goes wrong conceptually

A common misconception is that the tangent line slope is “the slope between two points that are very close.” That’s close, but not complete: the derivative is defined by a limit, not by “pick a small number.” AP questions will sometimes test whether you understand that the derivative is what happens as the interval shrinks to zero, not at some fixed small interval.

Exam Focus

• Typical question patterns:
  • Given a context (position, volume, temperature), compute average rate of change and interpret units.
  • Use a secant slope expression and discuss what happens as h \to 0.
  • From a graph, estimate the slope of the tangent line at a point.
• Common mistakes:
  • Mixing up f(a+h)-f(a) with f(a)-f(a+h) (sign errors).
  • Using h=0 directly in \frac{f(a+h)-f(a)}{h} (division by zero) instead of taking a limit.
  • Treating “tangent line touches once” as the definition (curves can cross their tangent lines).

The Limit Definition of the Derivative

The derivative is the mathematical way to define “instantaneous rate of change” and “tangent slope” precisely.

Derivative at a point

The derivative of f at x=a, if it exists, is defined by the limit:

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

Here, \frac{f(a+h)-f(a)}{h} is a secant slope (average rate of change) from a to a+h, and \lim_{h \to 0} means you shrink that interval down to a single point.

If the limit exists as a finite real number, then f is **differentiable** at a and the tangent slope is that limit.

Equivalent limit form

You may also see the definition written as:

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

This is the same idea: the point x approaches a. If you set x=a+h, then as h \to 0, x \to a, and the expressions match.

Worked example: derivative from the definition

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

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

Compute:

• f(a+h)=(a+h)^2=a^2+2ah+h^2
• f(a)=a^2

Substitute and simplify:

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

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

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

Cancel the factor of h (this is why you must not plug in h=0 too early):

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

Now evaluate:

f'(a)=2a

So the derivative function is f'(x)=2x.

Worked example: derivative at a particular point

Using the result above, the slope of the tangent line to y=x^2 at x=3 is:

f'(3)=2\cdot 3=6

What goes wrong algebraically

Most limit-definition errors come from:

• Expanding incorrectly, especially expressions like \left(a+h\right)^2.
• Canceling incorrectly: you can only cancel a factor of h after factoring it from every term in the numerator.
• Plugging in h=0 too soon: \frac{f(a+h)-f(a)}{h} is undefined at h=0, but the limit as h \to 0 can still exist.

Exam Focus

• Typical question patterns:
  • Use the limit definition to find f'(x) for a given polynomial or radical function.
  • Write an expression for f'(a) using limits (no simplification required).
  • Evaluate a derivative at a point using the definition (often to practice algebra/limits).
• Common mistakes:
  • Treating f'(a) as \frac{f(a)}{a}.
  • Forgetting the limit entirely and simplifying only the fraction.
  • Switching forms incorrectly (confusing x-a with a-x, causing sign errors).

Derivative Notation and What the Derivative Means

Once you have the definition, the next skill is fluency: recognizing different notations and interpreting what a derivative tells you.

Notation you must recognize

All of the following notations can represent “the derivative of y=f(x) with respect to x”:

You should also recognize second-derivative notation (even if its main uses come later):

Conceptually:

• f'(x) is a new function that outputs the slope at each input.
• f'(a) is a **number**: the slope (or instantaneous rate) at x=a.
• \frac{dy}{dx} emphasizes rates: “change in output per change in input.” Even though it looks like a fraction, treat it as a single piece of derivative notation.

Two core interpretations

1) Slope interpretation: If f describes a curve, then f'(a) is the slope of the tangent line to y=f(x) at x=a.

2) Rate-of-change interpretation: If f(x) is a quantity depending on x, then f'(a) is the instantaneous rate at which f changes when x=a.

Units: interpret them correctly

Units are a powerful sanity check. If f has units and x has units, then:

\text{units of } f'(x)=\frac{\text{units of } f}{\text{units of } x}

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

Local linearity and linearization

Differentiable functions are locally linear: near x=a, the graph behaves approximately like its tangent line.

The tangent line (linearization) at x=a is:

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

Worked example: interpreting meaning in context

Suppose V(t) is the volume of water in a tank (liters) at time t (minutes).

• If V'(7)=3, then at t=7 minutes, the volume is increasing at 3 liters per minute.
• If V'(7)=-3, then at t=7 minutes, the volume is decreasing at 3 liters per minute.

What goes wrong in interpretation

Students commonly confuse:

• f(a) vs. f'(a): one is the function value (height/amount), the other is the slope/rate.
• f'(a) vs. f'(x): one is a number, the other is a function.
• “Derivative is negative” does not mean the function is below the x-axis. It means the function is decreasing.

Exam Focus

• Typical question patterns:
  • Interpret f'(a) in context, including correct units.
  • Find the equation of the tangent line at x=a.
  • Explain the meaning of the sign of f'(a) for a real-world situation.
• Common mistakes:
  • Giving units of f instead of units of f'.
  • Using the point-slope formula with the wrong point (using \left(f(a),a\right) instead of \left(a,f(a)\right)).
  • Interpreting f'(a)=0 as “f(a)=0.”

Estimating and Approximating Derivatives from Tables and Graphs

In many AP problems, you aren’t given a formula for f. You might get a table of values, a graph, or a verbal description. The derivative is still defined the same way, but you estimate it by approximating the limiting secant slope.

Estimating f'(a) from a table

If you only know values of f at certain inputs, approximate the derivative using secant slopes with inputs very close to a.

• A right-hand approximation uses inputs x>a, such as a and a+h for small positive h.
• A left-hand approximation uses inputs x

A good estimate usually comes from comparing both sides:

• If left and right secant slopes are close, f'(a) is likely near that shared value.
• If they disagree significantly, f'(a) might not exist (or your data is too coarse).

Worked example: table estimate

Suppose a table gives:

• f(2)=5
• f(2.1)=5.23
• f(1.9)=4.79

Right-hand slope estimate at a=2:

\frac{f(2.1)-f(2)}{2.1-2}=\frac{5.23-5}{0.1}=2.3

Left-hand slope estimate at a=2:

\frac{f(2)-f(1.9)}{2-1.9}=\frac{5-4.79}{0.1}=2.1

So f'(2) is likely around 2.2.

Estimating slope from a graph

When you estimate f'(a) from a graph, you are estimating the slope of the tangent line.

A practical method:

1) At x=a, lightly sketch the tangent line (the line that best matches the curve’s direction there).
2) Pick two clear points on your tangent line (not necessarily on the curve).
3) Compute slope using rise over run.

When a graph suggests the derivative is zero

If the graph has a horizontal tangent at x=a, then:

f'(a)=0

This often happens at a smooth local maximum or minimum.

Connecting the sign of f'(x) to the behavior of f(x)

• If f'(x)>0, the function is increasing (at least locally).
• If f'(x)

What goes wrong with estimates

• Using two points on the curve near a gives a secant slope; it can be a decent estimate, but it’s still an approximation.
• On graphs, students may mistakenly choose points on the curve instead of on the tangent line they sketched.
• Rounding too early can distort the estimate.

Exam Focus

• Typical question patterns:
  • Estimate f'(a) from a table using symmetric points around a.
  • Estimate the slope of the tangent line from a graph and interpret its meaning.
  • Decide whether f'(a) is positive, negative, or zero from a graph.
• Common mistakes:
  • Using a large interval (points far from a) and calling it an instantaneous rate.
  • Mixing up left-hand and right-hand differences (sign errors in the slope).
  • Reporting f'(a) as a value of the function (confusing height with slope).

Differentiability and Continuity: When Does a Derivative Exist?

The derivative is defined using a limit, so the derivative exists only when that limit exists.

A key relationship to remember is:

• Differentiability implies continuity (at that point).
• Continuity does not necessarily imply differentiability.

What “differentiable at a” means

A function f is differentiable at x=a if the limit

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

exists as a finite number. Geometrically, that means the graph has a well-defined tangent slope at x=a.

Differentiable implies continuous

If f is differentiable at a, then f must also be continuous at a. You cannot have a derivative at a point where the function has a hole, jump, or asymptote.

A useful memory: differentiable is “stronger” than continuous.

Main ways differentiability can fail

Even if f is continuous, the derivative might fail to exist at a. Common causes (often tested visually) are:

1) Corner: left-hand and right-hand slopes approach different finite numbers.

2) Cusp: slopes approach infinity with opposite signs.

3) Vertical tangent: the tangent line is vertical, so the slope is infinite/undefined.

4) Discontinuity: if f is not continuous at a, it cannot be differentiable there.

One-sided derivatives

The right-hand derivative at a is:

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

The left-hand derivative at a is:

\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: a classic corner

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

From the right (small positive h):

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

From the left (small negative h):

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

Because the one-sided derivatives are not equal, f'(0) does not exist.

What goes wrong on AP-style questions

It’s not enough to say “not differentiable because it’s not smooth.” Strong AP explanations tie back to:

• left- and right-hand slopes not matching,
• the slope becoming infinite (vertical tangent), or
• the function not being continuous.

Exam Focus

• Typical question patterns:
  • From the graph, identify where f is not differentiable and explain why.
  • Given a piecewise function, determine where f'(a) exists (often using one-sided behavior).
  • True/false with justification: If f is differentiable at a, then f is continuous at a.
• Common mistakes:
  • Claiming continuity implies differentiability.
  • Saying “derivative does not exist because the function is not smooth” without specifying corner/cusp/vertical tangent/discontinuity.
  • Forgetting that a vertical tangent is not differentiable (slope undefined).

Fundamental Derivative Properties and Derivative Rules

Using the limit definition every time is tedious, so calculus develops rules that let you differentiate efficiently.

Linearity: constants, sums, and differences

1) Constant rule: if f(x)=k where k is a constant, then

f'(x)=0

Example: if f(x)=10 then f'(x)=0.

2) Constant multiple rule: if c is constant and f is differentiable, then

\frac{d}{dx}[cf(x)]=c f'(x)

3) Sum rule: if f and g are differentiable, then

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

4) Difference rule:

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

Derivative of x

If f(x)=x, then

f'(x)=1

The power rule

For positive integers n:

\frac{d}{dx}[x^n]=n x^{n-1}

A helpful description is “multiply down and decrease the power.” For example:

• x^4 becomes 4x^3
• 2x^2 becomes 4x

Polynomials (linearity + power rule)

Example:

f(x)=3x^4-2x^2+7x-5

Differentiate term-by-term:

f'(x)=12x^3-4x+7

Product rule

If f(x)=uv, then

f'(x)=u\frac{dv}{dx}+v\frac{du}{dx}

Mnemonic: “1d2 + 2d1” (first times derivative of second, plus second times derivative of first).

Example situation: differentiating \left(2x+7\right)\left(9x+8\right). You could multiply first, but the product rule is faster:

• Let u=2x+7 and v=9x+8
• Then \frac{du}{dx}=2 and \frac{dv}{dx}=9

So:

\frac{d}{dx}[\left(2x+7\right)\left(9x+8\right)]=(2x+7)\cdot 9+(9x+8)\cdot 2

which simplifies to:

36x+79

Quotient rule

If f(x)=\frac{u}{v}, then

f'(x)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}

Mnemonic: “low d high minus high d low over low squared.” (Denominator times derivative of numerator, minus numerator times derivative of denominator, all over denominator squared.)

Worked example: tangent line using derivative rules

Find the equation of the tangent line to f(x)=x^3-4x+1 at x=2.

Differentiate:

f'(x)=3x^2-4

Evaluate slope:

f'(2)=3\cdot 2^2-4=8

Find the point:

f(2)=2^3-4\cdot 2+1=1

Tangent line (point-slope form):

y-1=8(x-2)

Normal line (when defined)

If the tangent slope at x=a is f'(a) and f'(a) is nonzero, then the normal line slope is:

-\frac{1}{f'(a)}

Common misconceptions with rules

• Forgetting to multiply by the exponent in the power rule.
• Decreasing the exponent incorrectly.
• Mishandling negative coefficients (sign errors).
• Misusing the power rule on expressions like \left(3x+1\right)^5 (this requires the chain rule, which comes later).

Exam Focus

• Typical question patterns:
  • Differentiate a polynomial (or combination of functions) and evaluate f'(a).
  • Find an equation of the tangent line or normal line at a point.
  • Use the product rule or quotient rule to differentiate efficiently without expanding.
• Common mistakes:
  • Applying the power rule to a non-simple power (like a power of a linear expression) without chain rule.
  • Mixing up the order/signs in the quotient rule numerator.
  • Dropping terms incorrectly: constants disappear, but linear terms do not.

Derivatives of Common “Memory” Functions

Some derivatives are easier to memorize than to derive each time. In this unit, the most important are trig derivatives (sine and cosine), and it’s also helpful to know the basic exponential and logarithmic derivatives.

Trigonometric derivatives (sine and cosine)

Measured in radians (essential), the core facts are:

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

\frac{d}{dx}[\cos(x)]=-\sin(x)

Two important cautions:

1) These formulas assume x is in radians, not degrees.
2) The negative sign in the cosine derivative is crucial.

An intuitive check (unit circle / graph behavior): near x=0, \sin(x) increases most steeply when \cos(x) is near 1, and \cos(x) decreases most steeply when -\sin(x) is near 0 with the correct sign pattern.

Using linearity with trig derivatives

Example:

f(x)=3\sin(x)-2\cos(x)+x^2

Differentiate term-by-term:

f'(x)=3\cos(x)+2\sin(x)+2x

Notice the sign change:

\frac{d}{dx}[-2\cos(x)]=2\sin(x)

Worked example: slope of a trig-based model

Let h(t)=10\sin(t)+5 represent height (meters) over time t (seconds).

Differentiate:

h'(t)=10\cos(t)

Interpretation: h'(t) is vertical velocity in meters per second.

At t=0:

h'(0)=10\cos(0)=10

So at time 0, the height is increasing at 10 meters per second.

Exponential and logarithmic derivatives

These are also common “memory derivatives”:

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

For natural logarithms:

\frac{d}{dx}[\ln(x)]=\frac{1}{x}

(When working in real numbers, \ln(x) requires x>0.)

Common “memory derivative” mistakes

• Writing \frac{d}{dx}[\cos(x)]=\sin(x) (missing the negative).
• Mixing up which function you get back: derivative of sine is cosine; derivative of cosine is negative sine.
• Using degrees on a calculator and expecting the trig derivative relationships to hold numerically.

Exam Focus

• Typical question patterns:
  • Differentiate expressions mixing polynomials with \sin(x), \cos(x), e^x, and \ln(x).
  • Evaluate derivatives at special angles (in radians) or special inputs (like x=0).
  • Interpret a derivative as a rate (velocity, growth/decline, oscillation rate) with correct units.
• Common mistakes:
  • Dropping the negative sign in \frac{d}{dx}[\cos(x)].
  • Evaluating trig in degrees instead of radians.
  • Confusing function values like \sin(0) with derivative values like \cos(0).

Putting It Together: Tangent Lines, Rates, and Existence

Unit 2 problems often combine these ideas into multi-step reasoning:

• The derivative definition explains what the derivative is.
• Derivative rules let you compute it efficiently.
• Graphs and tables let you estimate it.
• Differentiability tells you when it exists.
• Tangent lines and units tell you how to interpret it.

Worked example: tangent line with interpretation

A function p(t) gives the population of a town (people) at time t (years):

p(t)=5000+120t-3t^2

Differentiate:

p'(t)=120-6t

Interpretation: p'(t) is the instantaneous rate of change of population in people per year.

At t=10:

p'(10)=60

So at year 10, population is increasing at 60 people per year.

Now find the tangent line at t=10. First the point:

p(10)=5900

Tangent line (linearization near t=10):

L(t)=5900+60(t-10)

Meaning: near year 10, population is approximately linear with slope 60 people per year.

Worked example: differentiability check from a graph description

If a graph is continuous but has a sharp corner at x=2 (left side slope about -1, right side slope about 3), then f(2) exists and the function is continuous there, but f'(2) does not exist because the left-hand and right-hand derivatives do not match.

What goes wrong when combining skills

• Computing a derivative correctly but interpreting it incorrectly (wrong units or wrong meaning).
• Finding a tangent slope correctly but using the wrong point in the line equation.
• Assuming differentiability because the function is continuous (the corner/cusp trap).

Exam Focus

• Typical question patterns:
  • Multi-part problems: compute f'(x), evaluate f'(a), write a tangent line, and interpret it.
  • Given a graph, identify where f'(x) is positive/negative/zero and where it does not exist.
  • Justify differentiability/continuity relationships in words.
• Common mistakes:
  • Writing a tangent line using f'(a) but forgetting to compute f(a).
  • Using derivative rules when the question explicitly requires the limit definition.
  • Giving an interpretation that describes f(a) (the amount) instead of f'(a) (the rate).