Defining and Differentiating Vector-Valued Functions
Differentiate vector components, separating vector velocity and acceleration from scalar speed.
College Board Topic 9.4 · CED p. 169 (PDF p. 174) · CED effective Fall 2020; project course outline 2026–27.
The vector notation packages the same parametric coordinates into one position vector. The derivative is componentwise: . This vector points in the instantaneous direction of motion when it is nonzero. Its magnitude is speed; its component ratio, where x′ is nonzero, is the tangent slope.
Differentiate a vector again component by component to get acceleration . A vector derivative is a vector, not a scalar. Velocity and speed are different outputs: velocity keeps direction, speed keeps magnitude.
You can read the same information from a table of derivative values. If and , the velocity is , speed is 5, and the tangent slope is −4/3. A table of positions would require a difference quotient to estimate these rates; it would not supply exact instantaneous derivatives.
Let's work through it
- For , obtain . Why: Differentiate each component separately.
- At , velocity is and speed is . Why: Evaluate first, then take the vector magnitude.
- Acceleration is , so . Why: Differentiate velocity componentwise.
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
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Retrieval 1
Worked answer and reason
Each component needs the chain-rule factor 2.
Retrieval 2
Worked answer and reason
Speed is the nonnegative vector magnitude.
Catch the tempting mistake
- A vector derivative is the derivative of the magnitude.
- Differentiate components to obtain a vector. Differentiating magnitude gives a scalar rate and can lose direction.
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Put it to work
- Unit 9 problem 10 · No calculator · analytic
- Unit 9 problem 11 · No calculator · analytic
- Unit 9 problem 12 · No calculator · numerical
For the teacher
- Have students translate coordinate, vector and verbal motion descriptions without recomputing derivatives.
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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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Acceleration in calculus
Acceleration is the derivative of velocity with respect to time. In one-dimensional motion, it is also the second derivative of position. For example, s(t)=t² meters gives constant acceleration 2 meters per second squared. A negative acceleration can increase speed when the velocity is also negative.
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Velocity in calculus
In one-dimensional motion, velocity is the derivative of position with respect to time. Its sign records direction relative to the chosen positive axis. For example, s(t)=t²−4t meters gives velocity 2t−4 meters per second. Speed is the magnitude of velocity and therefore cannot be negative.
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Difference quotient
A difference quotient divides a change in a function’s output by the associated nonzero input change. The form [f(a+h)−f(a)]/h is an average rate over an interval of length h. For example, for f(x)=x² it simplifies to 2a+h when h≠0. Taking its limit as h approaches 0 defines the derivative at a if the finite limit exists.