Calculus Examples

$r (t) = t^{2} \cdot i + (1 - t) \cdot j + (\frac{2}{3}) t^{3} \cdot k$

Step 1

By the Sum Rule, the derivative of $t^{2} \cdot i + (1 - t) \cdot j + (\frac{2}{3}) t^{3} \cdot k$ with respect to $t$ is $\frac{d}{d t} [t^{2} \cdot i] + \frac{d}{d t} [(1 - t) \cdot j] + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$ .

$\frac{d}{d t} [t^{2} \cdot i] + \frac{d}{d t} [(1 - t) \cdot j] + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 2

Evaluate $\frac{d}{d t} [t^{2} \cdot i]$ .

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Step 2.1

Since $i$ is constant with respect to $t$ , the derivative of $t^{2} i$ with respect to $t$ is $i \frac{d}{d t} [t^{2}]$ .

$i \frac{d}{d t} [t^{2}] + \frac{d}{d t} [(1 - t) \cdot j] + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$i (2 t) + \frac{d}{d t} [(1 - t) \cdot j] + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 2.3

Move $2$ to the left of $i$ .

$2 i t + \frac{d}{d t} [(1 - t) \cdot j] + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 3

Evaluate $\frac{d}{d t} [(1 - t) \cdot j]$ .

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Step 3.1

Since $j$ is constant with respect to $t$ , the derivative of $(1 - t) j$ with respect to $t$ is $j \frac{d}{d t} [1 - t]$ .

$2 i t + j \frac{d}{d t} [1 - t] + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 3.2

By the Sum Rule, the derivative of $1 - t$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [- t]$ .

$2 i t + j (\frac{d}{d t} [1] + \frac{d}{d t} [- t]) + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 3.3

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$2 i t + j (0 + \frac{d}{d t} [- t]) + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 3.4

Since $- 1$ is constant with respect to $t$ , the derivative of $- t$ with respect to $t$ is $- \frac{d}{d t} [t]$ .

$2 i t + j (0 - \frac{d}{d t} [t]) + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$2 i t + j (0 - 1 \cdot 1) + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 3.6

Multiply $- 1$ by $1$ .

$2 i t + j (0 - 1) + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 3.7

Subtract $1$ from $0$ .

$2 i t + j \cdot - 1 + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 3.8

Move $- 1$ to the left of $j$ .

$2 i t - 1 \cdot j + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 3.9

Rewrite $- 1 j$ as $- j$ .

$2 i t - j + \frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$

Step 4

Evaluate $\frac{d}{d t} [(\frac{2}{3}) t^{3} \cdot k]$ .

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Step 4.1

Combine $\frac{2}{3}$ and $t^{3}$ .

$2 i t - j + \frac{d}{d t} [\frac{2 t^{3}}{3} \cdot k]$

Step 4.2

Combine $\frac{2 t^{3}}{3}$ and $k$ .

$2 i t - j + \frac{d}{d t} [\frac{2 t^{3} k}{3}]$

Step 4.3

Since $\frac{2 k}{3}$ is constant with respect to $t$ , the derivative of $\frac{2 t^{3} k}{3}$ with respect to $t$ is $\frac{2 k}{3} \frac{d}{d t} [t^{3}]$ .

$2 i t - j + \frac{2 k}{3} \frac{d}{d t} [t^{3}]$

Step 4.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 3$ .

$2 i t - j + \frac{2 k}{3} (3 t^{2})$

Step 4.5

Combine $3$ and $\frac{2 k}{3}$ .

$2 i t - j + \frac{3 (2 k)}{3} t^{2}$

Step 4.6

Multiply $2$ by $3$ .

$2 i t - j + \frac{6 k}{3} t^{2}$

Step 4.7

Combine $\frac{6 k}{3}$ and $t^{2}$ .

$2 i t - j + \frac{6 k t^{2}}{3}$

Step 4.8

Cancel the common factor of $6$ and $3$ .

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Step 4.8.1

Factor $3$ out of $6 k t^{2}$ .

$2 i t - j + \frac{3 (2 k t^{2})}{3}$

Step 4.8.2

Cancel the common factors.

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Step 4.8.2.1

Factor $3$ out of $3$ .

$2 i t - j + \frac{3 (2 k t^{2})}{3 (1)}$

Step 4.8.2.2

Cancel the common factor.

$2 i t - j + \frac{3 (2 k t^{2})}{3 \cdot 1}$

Step 4.8.2.3

Rewrite the expression.

$2 i t - j + \frac{2 k t^{2}}{1}$

Step 4.8.2.4

Divide $2 k t^{2}$ by $1$ .

$2 i t - j + 2 k t^{2}$

Step 5

Reorder terms.

$2 i t + 2 t^{2} k - j$