Calculus Examples

$s (t) = \frac{6 t}{\sin (t)}$

Step 1

Since $6$ is constant with respect to $t$ , the derivative of $\frac{6 t}{\sin (t)}$ with respect to $t$ is $6 \frac{d}{d t} [\frac{t}{\sin (t)}]$ .

$6 \frac{d}{d t} [\frac{t}{\sin (t)}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = t$ and $g (t) = \sin (t)$ .

$6 \frac{\sin (t) \frac{d}{d t} [t] - t \frac{d}{d t} [\sin (t)]}{\sin^{2} (t)}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

$6 \frac{\sin (t) \cdot 1 - t \frac{d}{d t} [\sin (t)]}{\sin^{2} (t)}$

Step 3.2

Multiply $\sin (t)$ by $1$ .

$6 \frac{\sin (t) - t \frac{d}{d t} [\sin (t)]}{\sin^{2} (t)}$

Step 4

The derivative of $\sin (t)$ with respect to $t$ is $\cos (t)$ .

$6 \frac{\sin (t) - t \cos (t)}{\sin^{2} (t)}$

Step 5

Combine $6$ and $\frac{\sin (t) - t \cos (t)}{\sin^{2} (t)}$ .

$\frac{6 (\sin (t) - t \cos (t))}{\sin^{2} (t)}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{6 \sin (t) + 6 (- t \cos (t))}{\sin^{2} (t)}$

Step 6.2

Multiply $- 1$ by $6$ .

$\frac{6 \sin (t) - 6 t \cos (t)}{\sin^{2} (t)}$

Step 6.3

Reorder terms.

$\frac{- 6 t \cos (t) + 6 \sin (t)}{\sin^{2} (t)}$