Calculus Examples

$q (t) = 10 t \cos (0.5) + 10 t \sin (0.8)$

Step 1

By the Sum Rule, the derivative of $10 t \cos (0.5) + 10 t \sin (0.8)$ with respect to $t$ is $\frac{d}{d t} [10 t \cos (0.5)] + \frac{d}{d t} [10 t \sin (0.8)]$ .

$\frac{d}{d t} [10 t \cos (0.5)] + \frac{d}{d t} [10 t \sin (0.8)]$

Step 2

Evaluate $\frac{d}{d t} [10 t \cos (0.5)]$ .

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Step 2.1

Multiply $\cos (0.5)$ by $10$ .

$\frac{d}{d t} [8.77582561 t] + \frac{d}{d t} [10 t \sin (0.8)]$

Step 2.2

Since $8.77582561$ is constant with respect to $t$ , the derivative of $8.77582561 t$ with respect to $t$ is $8.77582561 \frac{d}{d t} [t]$ .

$8.77582561 \frac{d}{d t} [t] + \frac{d}{d t} [10 t \sin (0.8)]$

Step 2.3

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

$8.77582561 \cdot 1 + \frac{d}{d t} [10 t \sin (0.8)]$

Step 2.4

Multiply $8.77582561$ by $1$ .

$8.77582561 + \frac{d}{d t} [10 t \sin (0.8)]$

Step 3

Evaluate $\frac{d}{d t} [10 t \sin (0.8)]$ .

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Step 3.1

Multiply $\sin (0.8)$ by $10$ .

$8.77582561 + \frac{d}{d t} [7.1735609 t]$

Step 3.2

Since $7.1735609$ is constant with respect to $t$ , the derivative of $7.1735609 t$ with respect to $t$ is $7.1735609 \frac{d}{d t} [t]$ .

$8.77582561 + 7.1735609 \frac{d}{d t} [t]$

Step 3.3

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

$8.77582561 + 7.1735609 \cdot 1$

Step 3.4

Multiply $7.1735609$ by $1$ .

$8.77582561 + 7.1735609$

Step 4

Add $8.77582561$ and $7.1735609$ .

$15.94938652$