Calculus Examples

$\frac{1}{t^{3}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Multiply $3$ by $- 1$ .

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

Step 2

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

$- 3 t^{- 4}$

Step 3

Simplify.

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Step 3.1

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

$- 3 \frac{1}{t^{4}}$

Step 3.2

Combine terms.

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Step 3.2.1

Combine $- 3$ and $\frac{1}{t^{4}}$ .

$\frac{- 3}{t^{4}}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{3}{t^{4}}$