AP Calculus AB
AP Calculus AB · Lesson Review · September 2, 2026

The Derivative: Tangent Lines and Instantaneous Rate of Change

Turn nearby secant slopes into an exact tangent slope, derive a formula with a limit, and interpret the derivative as an instantaneous rate with meaningful units.

41-minute edited lesson12 chaptersCaptions + corrected transcript

01 · Watch

Lesson video

VTT captionsSRT captions
0:0001The Derivative: Tangent Lines and Instantaneous Rate0:3802Reviewing Secant-Line Slope4:0203The Tangent-Line Goal6:1104Using a Limit to Find the Slope10:2905Two Definitions of the Derivative14:0406Deriving a Polynomial Formula18:4407A Function of Tangent Slopes21:5508Meaning and Notation24:4109Radical Function and the Conjugate30:5310Equation of the Tangent Line32:2311When a Derivative Does Not Exist33:3812Instantaneous Rate of Flow

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02 · Understand

Learning targets

03 · Vocabulary

Key definitions

Keep the geometry, algebra, and units connected throughout every problem.

01Secant line
A line through two points on a graph. Its slope is an average rate of change: msec=f(b)−f(a)b−am_{\mathrm{sec}}=\frac{f(b)-f(a)}{b-a}.
02Tangent line
The line that locally matches the graph's direction at one point. Its slope is the derivative at that input.
03Derivative at a point
The limiting value of nearby secant slopes: f′(a)=lim⁡x→af(x)−f(a)x−af'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.
04Derivative function
A function that returns the tangent-line slope at each input: f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}.
05Instantaneous rate of change
The derivative interpreted in context, with units equal to output units divided by input units.

04 · Apply

Representative examples

Every definition problem follows the same arc: substitute, simplify the indeterminate form, cancel the factor that approaches zero, then evaluate.

Secant to tangent

A point-specific derivative

  1. Let f(x)=x2−2x−1f(x)=x^2-2x-1 and find the tangent slope at x=3x=3.
  2. f′(3)=lim⁡x→3f(x)−f(3)x−3f'(3)=\lim_{x\to3}\frac{f(x)-f(3)}{x-3}
  3. Factor and cancel the common factor x−3x-3, then substitute.
  4. f′(3)=4f'(3)=4, so the tangent line through (3,2)(3,2) is y−2=4(x−3)y-2=4(x-3).
Derivative function

Use the h-definition on a polynomial

  1. f′(x)=lim⁡h→0(x+h)2−2(x+h)−1−(x2−2x−1)hf'(x)=\lim_{h\to0}\frac{(x+h)^2-2(x+h)-1-(x^2-2x-1)}{h}
  2. Expand and combine like terms: 2xh+h2−2hh\frac{2xh+h^2-2h}{h}.
  3. Factor and cancel hh: lim⁡h→0(2x+h−2)\lim_{h\to0}(2x+h-2).
  4. f′(x)=2x−2f'(x)=2x-2.
Radical

Use a conjugate

  1. For g(x)=x+1g(x)=\sqrt{x+1}, substitute into the h-definition.
  2. Multiply by the conjugate x+h+1+x+1\sqrt{x+h+1}+\sqrt{x+1}.
  3. After cancellation, g′(x)=12x+1g'(x)=\frac{1}{2\sqrt{x+1}}.
  4. At x=8x=8, g′(8)=16g'(8)=\frac16 and y−3=16(x−8)y-3=\frac16(x-8).
Application

Interpret an instantaneous flow rate

  1. V(t)=−(t−1)2+5V(t)=-(t-1)^2+5 measures water volume in gallons after tt hours.
  2. The limit definition simplifies to V′(t)=−2t+2V'(t)=-2t+2.
  3. V′(3)=−4 gallons per hourV'(3)=-4\ \text{gallons per hour}
  4. The negative sign means the volume is decreasing at 44 gallons per hour.

05 · Avoid

Common mistakes

Confusing secant and tangent slopes.

A secant uses two distinct points; the tangent slope is their limiting value as the points come together.

Substituting too soon.

The difference quotient initially gives 0/00/0. Simplify before evaluating the limit.

Forgetting parentheses.

In f(x+h)−f(x)f(x+h)-f(x), subtract the entire second function expression.

Canceling terms instead of factors.

Factor hh from the complete numerator before canceling it with the denominator.

Losing the point.

A tangent equation needs both the derivative value and the point (a,f(a))(a,f(a)).

Dropping units or meaning.

State output units per input unit and interpret whether the quantity increases or decreases.

06 · Check yourself

Quick check

Try each question before opening its answer.

1What geometric quantity does f′(a)f'(a) represent?

The slope of the tangent line to y=f(x)y=f(x) at x=ax=a.

2Why does the h-definition produce a function instead of one number?

The variable xx remains after hh approaches zero, so the result returns a tangent slope for every allowed input.

3If g′(8)=16g'(8)=\frac16 and g(8)=3g(8)=3, what is the tangent line?

y−3=16(x−8)y-3=\frac16(x-8).

4What does V′(3)=−4 gal/hrV'(3)=-4\ \text{gal/hr} mean?

At three hours, the water volume is decreasing at 4 gallons per hour.

07 · Revisit

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