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

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

Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

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

open angle brackets x open parentheses t close parentheses comma space y open parentheses t close parentheses close angle brackets apostrophe space equals open angle brackets x apostrophe open parentheses t close parentheses comma space y apostrophe open parentheses t close parentheses close angle brackets

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

Worked Example

A vector-valued function is given by open angle brackets 3 t squared comma space 5 e to the power of t plus sin space t close angle brackets.

Find its second derivative when t equals 0.

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

Start by differentiating both components once

open angle brackets 6 t comma space 5 e to the power of t plus cos space t close angle brackets

Then differentiate both components again

open angle brackets 6 comma space 5 e to the power of t minus sin space t close angle brackets

Substitute in t equals 0 and simplify

open angle brackets 6 comma space 5 e to the power of 0 minus sin space 0 close angle brackets

It is important to give your final answer in vector notation

open angle brackets 6 comma space 5 close angle brackets

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    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.