On this page we explain how to take derivatives of vector functions.
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Derivatives of vector functions require special techniques. This video clip shows some good examples and explains derivatives well.
video by Dr Chris Tisdell 

Try your hand at these practice problems.
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 

\(\displaystyle{ \vec{r}'(t) = [\pi\sin(\pi t)]\hat{i} + \left[\frac{e^t(t2)}{t^3}\right]\hat{j} + 12t^2\hat{k}}\)
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
video by PatrickJMT 

Final Answer
\(\displaystyle{ \vec{r}'(t) = [\pi\sin(\pi t)]\hat{i} + \left[\frac{e^t(t2)}{t^3}\right]\hat{j} + 12t^2\hat{k}}\)
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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
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 

\( \vec{r}'(t) = \langle \sin t+t\cos t, 2t, \cos 2t2t\sin 2t \rangle \)
Problem Statement
Find the derivative of \( \vec{r}(t) = \langle t\sin t, t^2, t\cos 2t\rangle \).
Solution
video by Krista King Math 

Final Answer
\( \vec{r}'(t) = \langle \sin t+t\cos t, 2t, \cos 2t2t\sin 2t \rangle \)
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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
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
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
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.