Calculus Examples

$\frac{1}{s} - 10 \csc (s)$

Step 1

By the Sum Rule, the derivative of $\frac{1}{s} - 10 \csc (s)$ with respect to $s$ is $\frac{d}{d s} [\frac{1}{s}] + \frac{d}{d s} [- 10 \csc (s)]$ .

$\frac{d}{d s} [\frac{1}{s}] + \frac{d}{d s} [- 10 \csc (s)]$

Step 2

Evaluate $\frac{d}{d s} [\frac{1}{s}]$ .

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Step 2.1

Rewrite $\frac{1}{s}$ as $s^{- 1}$ .

$\frac{d}{d s} [s^{- 1}] + \frac{d}{d s} [- 10 \csc (s)]$

Step 2.2

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

$- s^{- 2} + \frac{d}{d s} [- 10 \csc (s)]$

Step 3

Evaluate $\frac{d}{d s} [- 10 \csc (s)]$ .

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Step 3.1

Since $- 10$ is constant with respect to $s$ , the derivative of $- 10 \csc (s)$ with respect to $s$ is $- 10 \frac{d}{d s} [\csc (s)]$ .

$- s^{- 2} - 10 \frac{d}{d s} [\csc (s)]$

Step 3.2

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

$- s^{- 2} - 10 (- \csc (s) \cot (s))$

Step 3.3

Multiply $- 1$ by $- 10$ .

$- s^{- 2} + 10 \csc (s) \cot (s)$

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{s^{2}} + 10 \csc (s) \cot (s)$

Step 4.2

Reorder terms.

$10 \cot (s) \csc (s) - \frac{1}{s^{2}}$