Defining the Derivative of a Function and Using Derivative Notation
Connect the derivative function, derivative notation, and a tangent line at a specified point.
College Board Topic 2.2 · CED p. 56 (PDF p. 61) · CED effective Fall 2020; project course outline 2026–27.
Let the base input vary. The rule f′(x)=lim(h→0)[f(x+h)−f(x)]/h defines a new function on the inputs where that finite limit exists. f′(a) is one slope; f′ is the collection of slopes.
For y=f(x), the notations f′(x), y′ and dy/dx describe the first derivative. dy/dx is derivative notation here; reading it as an ordinary numerical quotient at one point can mislead. Graphically a derivative is a tangent slope, numerically it can be estimated from close secants, analytically it is a limiting quotient, and verbally it is an instantaneous rate.
A tangent line at (a,f(a)) has equation y−f(a)=f′(a)(x−a) when the finite derivative exists. Its slope belongs to f′; its point belongs to f. For f(x)=x², the derivative function is 2x, while the tangent at a=3 is y−9=6(x−3). A vertical tangent has no finite slope and is treated separately.
Let's work through it
- Expand [(x+h)²−x²]/h to 2x+h for h≠0. Reason: the derivative definition uses equal nearby expressions.
- Take h→0 to obtain f′(x)=2x. Reason: the limiting quotient is finite for every x.
- At x=3, use the point (3,9) and slope 6 to write y−9=6(x−3). Reason: point-slope form uses the function value and its derivative at the same input.
Try it from memory
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Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
If f(2)=5 and f′(2)=−3, write the tangent line.
See every recall question with its answer
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Retrieval 1
If f(2)=5 and f′(2)=−3, write the tangent line.
Worked answer and reason
y−5=−3(x−2).
The slope is −3 and the tangent point is (2,5).
Retrieval 2
For y=f(t), what notation records instantaneous change in y per unit t?
Worked answer and reason
dy/dt or f′(t).
The derivative notation follows the input variable.
Catch the tempting mistake
- The derivative f′(a) is the height of f at a.
- It is the tangent slope or instantaneous rate; the height is f(a).
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 2 problem 4 · No calculator · analytic
- Unit 2 problem 5 · No calculator · analytic
- Unit 2 problem 6 · No calculator · graphical
- Unit 2 problem 7 · No calculator · verbal
- Unit 2 problem 22 · No calculator · analytic
For the teacher
- Make students label the point and slope before writing a tangent equation.
- Compare a graph of f with a graph of its slopes; shared input does not mean shared output.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
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Instantaneous rate of change
An instantaneous rate of change is the limiting average rate around one input when that limit exists. For a differentiable function, it is its derivative at that input. For example, position s(t)=t² meters has instantaneous rate 4 meters per second at t=2 seconds. The rate’s units are output units per input unit.
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Vertical tangent
A vertical tangent is a tangent line parallel to the vertical axis, with no finite slope. A graph can have a vertical tangent even though its derivative does not exist as a finite real number at that input. For example, y=∛x has the vertical tangent x=0 at the origin as its nearby slopes tend to positive infinity. This case is separate from the finite-slope tangent approximation available at a differentiable point.