Calculus Examples

$V = \cos (102 t) \sin (201 t)$ , $t = 4$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = \cos (102 t)$ and $g (t) = \sin (201 t)$ .

$\cos (102 t) \frac{d}{d t} [\sin (201 t)] + \sin (201 t) \frac{d}{d t} [\cos (102 t)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \sin (t)$ and $g (t) = 201 t$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $201 t$ .

$\cos (102 t) (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d t} [201 t]) + \sin (201 t) \frac{d}{d t} [\cos (102 t)]$

Step 2.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (102 t) (\cos (u_{1}) \frac{d}{d t} [201 t]) + \sin (201 t) \frac{d}{d t} [\cos (102 t)]$

Step 2.3

Replace all occurrences of $u_{1}$ with $201 t$ .

$\cos (102 t) (\cos (201 t) \frac{d}{d t} [201 t]) + \sin (201 t) \frac{d}{d t} [\cos (102 t)]$

Step 3

Differentiate.

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Step 3.1

Since $201$ is constant with respect to $t$ , the derivative of $201 t$ with respect to $t$ is $201 \frac{d}{d t} [t]$ .

$\cos (102 t) \cos (201 t) (201 \frac{d}{d t} [t]) + \sin (201 t) \frac{d}{d t} [\cos (102 t)]$

Step 3.2

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

$\cos (102 t) \cos (201 t) (201 \cdot 1) + \sin (201 t) \frac{d}{d t} [\cos (102 t)]$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $201$ by $1$ .

$\cos (102 t) \cos (201 t) \cdot 201 + \sin (201 t) \frac{d}{d t} [\cos (102 t)]$

Step 3.3.2

Move $201$ to the left of $\cos (102 t) \cos (201 t)$ .

$201 \cdot (\cos (102 t) \cos (201 t)) + \sin (201 t) \frac{d}{d t} [\cos (102 t)]$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cos (t)$ and $g (t) = 102 t$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $102 t$ .

$201 \cos (102 t) \cos (201 t) + \sin (201 t) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d t} [102 t])$

Step 4.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$201 \cos (102 t) \cos (201 t) + \sin (201 t) (- \sin (u_{2}) \frac{d}{d t} [102 t])$

Step 4.3

Replace all occurrences of $u_{2}$ with $102 t$ .

$201 \cos (102 t) \cos (201 t) + \sin (201 t) (- \sin (102 t) \frac{d}{d t} [102 t])$

Step 5

Differentiate.

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Step 5.1

Since $102$ is constant with respect to $t$ , the derivative of $102 t$ with respect to $t$ is $102 \frac{d}{d t} [t]$ .

$201 \cos (102 t) \cos (201 t) + \sin (201 t) (- \sin (102 t) (102 \frac{d}{d t} [t]))$

Step 5.2

Multiply $102$ by $- 1$ .

$201 \cos (102 t) \cos (201 t) + \sin (201 t) (- 102 \sin (102 t) \frac{d}{d t} [t])$

Step 5.3

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

$201 \cos (102 t) \cos (201 t) + \sin (201 t) (- 102 \sin (102 t) \cdot 1)$

Step 5.4

Simplify the expression.

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Step 5.4.1

Multiply $- 102$ by $1$ .

$201 \cos (102 t) \cos (201 t) + \sin (201 t) (- 102 \sin (102 t))$

Step 5.4.2

Reorder terms.

$201 \cos (102 t) \cos (201 t) - 102 \sin (102 t) \sin (201 t)$

Step 6

Evaluate the derivative at $t = 4$ .

$201 \cos (102 (4)) \cos (201 (4)) - 102 \sin (102 (4)) \sin (201 (4))$

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $102$ by $4$ .

$201 \cos (408) \cos (201 (4)) - 102 \sin (102 (4)) \sin (201 (4))$

Step 7.1.2

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$201 \cos (48) \cos (201 (4)) - 102 \sin (102 (4)) \sin (201 (4))$

Step 7.1.3

Evaluate $\cos (48)$ .

$201 \cdot 0.6691306 \cos (201 (4)) - 102 \sin (102 (4)) \sin (201 (4))$

Step 7.1.4

Multiply $201$ by $0.6691306$ .

$134.49525187 \cos (201 (4)) - 102 \sin (102 (4)) \sin (201 (4))$

Step 7.1.5

Multiply $201$ by $4$ .

$134.49525187 \cos (804) - 102 \sin (102 (4)) \sin (201 (4))$

Step 7.1.6

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$134.49525187 \cos (84) - 102 \sin (102 (4)) \sin (201 (4))$

Step 7.1.7

Evaluate $\cos (84)$ .

$134.49525187 \cdot 0.10452846 - 102 \sin (102 (4)) \sin (201 (4))$

Step 7.1.8

Multiply $134.49525187$ by $0.10452846$ .

$14.05858199 - 102 \sin (102 (4)) \sin (201 (4))$

Step 7.1.9

Multiply $102$ by $4$ .

$14.05858199 - 102 \sin (408) \sin (201 (4))$

Step 7.1.10

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$14.05858199 - 102 \sin (48) \sin (201 (4))$

Step 7.1.11

Evaluate $\sin (48)$ .

$14.05858199 - 102 \cdot 0.74314482 \sin (201 (4))$

Step 7.1.12

Multiply $- 102$ by $0.74314482$ .

$14.05858199 - 75.80077219 \sin (201 (4))$

Step 7.1.13

Multiply $201$ by $4$ .

$14.05858199 - 75.80077219 \sin (804)$

Step 7.1.14

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$14.05858199 - 75.80077219 \sin (84)$

Step 7.1.15

Evaluate $\sin (84)$ .

$14.05858199 - 75.80077219 \cdot 0.99452189$

Step 7.1.16

Multiply $- 75.80077219$ by $0.99452189$ .

$14.05858199 - 75.38552763$

Step 7.2

Subtract $75.38552763$ from $14.05858199$ .

$- 61.32694564$