## 17Calculus - Derivatives of Vector Functions

##### 17Calculus

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.

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

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

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

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

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

Problem Statement

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

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

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

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

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

Problem Statement

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

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

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

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

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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 $$\vec{r}'(1)$$.

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

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

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

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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 $$\vec{r}'(\pi/2)$$.

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

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

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

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

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

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

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

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