Calculus Examples

$w = \frac{3}{t^{3}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (w) = \frac{d}{d t} (\frac{3}{t^{3}})$

Step 2

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

$w'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.3

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

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

Step 3.4

Multiply $- 3$ by $3$ .

$- 9 t^{- 4}$

Step 3.5

Simplify.

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Step 3.5.1

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

$- 9 \frac{1}{t^{4}}$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Combine $- 9$ and $\frac{1}{t^{4}}$ .

$\frac{- 9}{t^{4}}$

Step 3.5.2.2

Move the negative in front of the fraction.

$- \frac{9}{t^{4}}$

Step 4

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

$w' = - \frac{9}{t^{4}}$

Step 5

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

$\frac{d w}{d t} = - \frac{9}{t^{4}}$