Calculus Examples

$s = t^{5} - \csc (t) + 18$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (t^{5} - \csc (t) + 18)$

Step 2

The derivative of $s$ with respect to $t$ is $s'$ .

$s'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $t^{5} - \csc (t) + 18$ with respect to $t$ is $\frac{d}{d t} [t^{5}] + \frac{d}{d t} [- \csc (t)] + \frac{d}{d t} [18]$ .

$\frac{d}{d t} [t^{5}] + \frac{d}{d t} [- \csc (t)] + \frac{d}{d t} [18]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 5$ .

$5 t^{4} + \frac{d}{d t} [- \csc (t)] + \frac{d}{d t} [18]$

Step 3.2

Evaluate $\frac{d}{d t} [- \csc (t)]$ .

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Step 3.2.1

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

$5 t^{4} - \frac{d}{d t} [\csc (t)] + \frac{d}{d t} [18]$

Step 3.2.2

The derivative of $\csc (t)$ with respect to $t$ is $- \csc (t) \cot (t)$ .

$5 t^{4} - (- \csc (t) \cot (t)) + \frac{d}{d t} [18]$

Step 3.2.3

Multiply $- 1$ by $- 1$ .

$5 t^{4} + 1 (\csc (t) \cot (t)) + \frac{d}{d t} [18]$

Step 3.2.4

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

$5 t^{4} + \csc (t) \cot (t) + \frac{d}{d t} [18]$

Step 3.3

Since $18$ is constant with respect to $t$ , the derivative of $18$ with respect to $t$ is $0$ .

$5 t^{4} + \csc (t) \cot (t) + 0$

Step 3.4

Simplify.

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Step 3.4.1

Add $5 t^{4} + \csc (t) \cot (t)$ and $0$ .

$5 t^{4} + \csc (t) \cot (t)$

Step 3.4.2

Reorder terms.

$5 t^{4} + \cot (t) \csc (t)$

Step 4

Reform the equation by setting the left side equal to the right side.

$s' = 5 t^{4} + \cot (t) \csc (t)$

Step 5

Replace $s'$ with $\frac{d s}{d t}$ .

$\frac{d s}{d t} = 5 t^{4} + \cot (t) \csc (t)$