AP®MathsCollege BoardCalculus BCRevision NotesUnit 9: Parametric Equations, Vector-Valued Functions & Polar CoordinatesVector-Valued FunctionsDerivatives of Vector-Valued Functions

Derivatives of Vector-Valued Functions (College Board AP® Calculus BC): Revision Note

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on 19 May 2026

Derivatives of vector-valued functions

How do I find derivatives of vector-valued functions?

The derivative of a vector-valued function is a vector-valued function where both components have been differentiated separately

$\frac{d r}{d t} = < x' (t), y' (t) >$

You can keep differentiating vector-valued functions to find higher-order derivatives

Worked Example

A vector-valued function is given by $< 3 t^{2}, 5 e^{t} + \sin t >$ .

Find its second derivative with respect to $t$ at the point when $t = 0$ .

Answer:

To find the second derivative, you need to differentiate both components inside the vector twice

Start by differentiating both components once

$< 6 t, 5 e^{t} + \cos t >$

Then differentiate both components again

$< 6, 5 e^{t} - \sin t >$

Substitute in $t = 0$ and simplify

$< 6, 5 e^{0} - \sin 0 >$

It is important to give your final answer in vector notation

$r' (0) = < 6, 5 >$

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