Calculus Examples

$g (t) = \frac{2}{t^{3}}$

Step 1

Since $2$ is constant with respect to $t$ , the derivative of $\frac{2}{t^{3}}$ with respect to $t$ is $2 \frac{d}{d t} [\frac{1}{t^{3}}]$ .

$2 \frac{d}{d t} [\frac{1}{t^{3}}]$

Step 2

Apply basic rules of exponents.

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Step 2.1

Rewrite $\frac{1}{t^{3}}$ as ${(t^{3})}^{- 1}$ .

$2 \frac{d}{d t} [{(t^{3})}^{- 1}]$

Step 2.2

Multiply the exponents in ${(t^{3})}^{- 1}$ .

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Step 2.2.1

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

$2 \frac{d}{d t} [t^{3 \cdot - 1}]$

Step 2.2.2

Multiply $3$ by $- 1$ .

$2 \frac{d}{d t} [t^{- 3}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = - 3$ .

$2 (- 3 t^{- 4})$

Step 4

Multiply $- 3$ by $2$ .

$- 6 t^{- 4}$

Step 5

Simplify.

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Step 5.1

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

$- 6 \frac{1}{t^{4}}$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $- 6$ and $\frac{1}{t^{4}}$ .

$\frac{- 6}{t^{4}}$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{6}{t^{4}}$