\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - Derivatives of Vector Functions

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

On this page we explain how to take derivatives of vector functions.

Derivatives of vector functions require special techniques. This video clip shows some good examples and explains derivatives well.

Dr Chris Tisdell - derivatives of vector functions [13mins-36secs]

video by Dr Chris Tisdell

Try your hand at these practice problems.

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Practice

Unless otherwise instructed, calculate the derivative of these vector functions. If a value is given, also calculate the derivative at that value.

\(\displaystyle{ \vec{r}(t) = [ \cos(\pi t)]\hat{i} + \left[ \frac{e^t}{t^2} \right]\hat{j} + 4t^3\hat{k} }\)

Problem Statement

Find the derivative of \(\displaystyle{ \vec{r}(t) = [ \cos(\pi t)]\hat{i} + \left[ \frac{e^t}{t^2} \right]\hat{j} + 4t^3\hat{k} }\)

Final Answer

\(\displaystyle{ \vec{r}'(t) = [-\pi\sin(\pi t)]\hat{i} + \left[\frac{e^t(t-2)}{t^3}\right]\hat{j} + 12t^2\hat{k}}\)

Problem Statement

Find the derivative of \(\displaystyle{ \vec{r}(t) = [ \cos(\pi t)]\hat{i} + \left[ \frac{e^t}{t^2} \right]\hat{j} + 4t^3\hat{k} }\)

Solution

PatrickJMT - 703 video solution

video by PatrickJMT

Final Answer

\(\displaystyle{ \vec{r}'(t) = [-\pi\sin(\pi t)]\hat{i} + \left[\frac{e^t(t-2)}{t^3}\right]\hat{j} + 12t^2\hat{k}}\)

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\( \vec{r}(t) = e^{t^2}\vhati - \vhatj + \ln(1+3t)\vhatk \)

Problem Statement

Find the derivative of \( \vec{r}(t) = e^{t^2}\vhati - \vhatj + \ln(1+3t)\vhatk \).

Final Answer

\( \vec{r}’(t) = 2te^{t^2}\vhati + 3/(1+3t)\vhatk \)

Problem Statement

Find the derivative of \( \vec{r}(t) = e^{t^2}\vhati - \vhatj + \ln(1+3t)\vhatk \).

Solution

Krista King Math - 2027 video solution

video by Krista King Math

Final Answer

\( \vec{r}’(t) = 2te^{t^2}\vhati + 3/(1+3t)\vhatk \)

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\( \vec{r}(t) = \langle t\sin t, t^2, t\cos 2t\rangle \)

Problem Statement

Find the derivative of \( \vec{r}(t) = \langle t\sin t, t^2, t\cos 2t\rangle \).

Final Answer

\( \vec{r}'(t) = \langle \sin t+t\cos t, 2t, \cos 2t-2t\sin 2t \rangle \)

Problem Statement

Find the derivative of \( \vec{r}(t) = \langle t\sin t, t^2, t\cos 2t\rangle \).

Solution

Krista King Math - 2028 video solution

video by Krista King Math

Final Answer

\( \vec{r}'(t) = \langle \sin t+t\cos t, 2t, \cos 2t-2t\sin 2t \rangle \)

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\(\displaystyle{ \vec{r}(t) = \frac{5}{t^2}\vhat{i} - 4\sqrt{t}\vhat{j} }\) and \( \vec{r}'(1) \)

Problem Statement

Find the derivative of \(\displaystyle{ \vec{r}(t) = \frac{5}{t^2}\vhat{i} - 4\sqrt{t}\vhat{j} }\) and \( \vec{r}'(1) \).

Final Answer

\( \vec{r}'(t) = (-10/t^3)\vhat{i} - (2/\sqrt{t})\vhat{j} \)
\( \vec{r}'(1) = -10\vhat{i} - 2\vhat{j} \)

Problem Statement

Find the derivative of \(\displaystyle{ \vec{r}(t) = \frac{5}{t^2}\vhat{i} - 4\sqrt{t}\vhat{j} }\) and \( \vec{r}'(1) \).

Solution

MIP4U - 2034 video solution

video by MIP4U

Final Answer

\( \vec{r}'(t) = (-10/t^3)\vhat{i} - (2/\sqrt{t})\vhat{j} \)
\( \vec{r}'(1) = -10\vhat{i} - 2\vhat{j} \)

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\( \vec{r}(t) = 3\cos(t)\vhat{i} + 2\sin(t)\vhat{j} - t^2\vhat{k} \) and \(\vec{r}'(\pi/2)\)

Problem Statement

Find the derivative of \( \vec{r}(t) = 3\cos(t)\vhat{i} + 2\sin(t)\vhat{j} - t^2\vhat{k} \) and \(\vec{r}'(\pi/2)\).

Final Answer

\( \vec{r}'(t) = -3\sin(t)\vhat{i} + 2\cos(t)\vhat{j} - 2t\vhat{k} \)
\( \vec{r}'(\pi/2) = -3\vhat{i} - \pi\vhat{k} \)

Problem Statement

Find the derivative of \( \vec{r}(t) = 3\cos(t)\vhat{i} + 2\sin(t)\vhat{j} - t^2\vhat{k} \) and \(\vec{r}'(\pi/2)\).

Solution

MIP4U - 2035 video solution

video by MIP4U

Final Answer

\( \vec{r}'(t) = -3\sin(t)\vhat{i} + 2\cos(t)\vhat{j} - 2t\vhat{k} \)
\( \vec{r}'(\pi/2) = -3\vhat{i} - \pi\vhat{k} \)

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\( \vec{r}(t) = 3\tan(7t)\vhat{i} + \sin^2(t)\vhat{j} - 4\ln t\vhat{k} \)

Problem Statement

Find the derivative of \( \vec{r}(t) = 3\tan(7t)\vhat{i} + \sin^2(t)\vhat{j} - 4\ln t\vhat{k} \).

Final Answer

\( \vec{r}'(t) = 21\sec^2(7t)\vhat{i} + 2\sin(t)\cos(t)\vhat{j} - (4/t)\vhat{k} \)

Problem Statement

Find the derivative of \( \vec{r}(t) = 3\tan(7t)\vhat{i} + \sin^2(t)\vhat{j} - 4\ln t\vhat{k} \).

Solution

MIP4U - 2036 video solution

video by MIP4U

Final Answer

\( \vec{r}'(t) = 21\sec^2(7t)\vhat{i} + 2\sin(t)\cos(t)\vhat{j} - (4/t)\vhat{k} \)

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Practice Instructions

Unless otherwise instructed, calculate the derivative of these vector functions. If a value is given, also calculate the derivative at that value.

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