Calculus Examples

$A (s) = - \frac{12}{s^{5}}$

Step 1

Since $- 12$ is constant with respect to $s$ , the derivative of $- \frac{12}{s^{5}}$ with respect to $s$ is $- 12 \frac{d}{d s} [\frac{1}{s^{5}}]$ .

$- 12 \frac{d}{d s} [\frac{1}{s^{5}}]$

Step 2

Apply basic rules of exponents.

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Step 2.1

Rewrite $\frac{1}{s^{5}}$ as ${(s^{5})}^{- 1}$ .

$- 12 \frac{d}{d s} [{(s^{5})}^{- 1}]$

Step 2.2

Multiply the exponents in ${(s^{5})}^{- 1}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 12 \frac{d}{d s} [s^{5 \cdot - 1}]$

Step 2.2.2

Multiply $5$ by $- 1$ .

$- 12 \frac{d}{d s} [s^{- 5}]$

Step 3

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

$- 12 (- 5 s^{- 6})$

Step 4

Multiply $- 5$ by $- 12$ .

$60 s^{- 6}$

Step 5

Simplify.

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Step 5.1

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

$60 \frac{1}{s^{6}}$

Step 5.2

Combine $60$ and $\frac{1}{s^{6}}$ .

$\frac{60}{s^{6}}$