Calculus Examples

$\sqrt[3]{t} (8 - t)$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{t}$ as $t^{\frac{1}{3}}$ .

$\frac{d}{d t} [t^{\frac{1}{3}} (8 - t)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t^{\frac{1}{3}}$ and $g (t) = 8 - t$ .

$t^{\frac{1}{3}} \frac{d}{d t} [8 - t] + (8 - t) \frac{d}{d t} [t^{\frac{1}{3}}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $8 - t$ with respect to $t$ is $\frac{d}{d t} [8] + \frac{d}{d t} [- t]$ .

$t^{\frac{1}{3}} (\frac{d}{d t} [8] + \frac{d}{d t} [- t]) + (8 - t) \frac{d}{d t} [t^{\frac{1}{3}}]$

Step 3.2

Since $8$ is constant with respect to $t$ , the derivative of $8$ with respect to $t$ is $0$ .

$t^{\frac{1}{3}} (0 + \frac{d}{d t} [- t]) + (8 - t) \frac{d}{d t} [t^{\frac{1}{3}}]$

Step 3.3

Add $0$ and $\frac{d}{d t} [- t]$ .

$t^{\frac{1}{3}} \frac{d}{d t} [- t] + (8 - t) \frac{d}{d t} [t^{\frac{1}{3}}]$

Step 3.4

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

$t^{\frac{1}{3}} (- \frac{d}{d t} [t]) + (8 - t) \frac{d}{d t} [t^{\frac{1}{3}}]$

Step 3.5

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

$t^{\frac{1}{3}} (- 1 \cdot 1) + (8 - t) \frac{d}{d t} [t^{\frac{1}{3}}]$

Step 3.6

Simplify the expression.

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Step 3.6.1

Multiply $- 1$ by $1$ .

$t^{\frac{1}{3}} \cdot - 1 + (8 - t) \frac{d}{d t} [t^{\frac{1}{3}}]$

Step 3.6.2

Move $- 1$ to the left of $t^{\frac{1}{3}}$ .

$- 1 \cdot t^{\frac{1}{3}} + (8 - t) \frac{d}{d t} [t^{\frac{1}{3}}]$

Step 3.6.3

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

$- t^{\frac{1}{3}} + (8 - t) \frac{d}{d t} [t^{\frac{1}{3}}]$

Step 3.7

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

$- t^{\frac{1}{3}} + (8 - t) (\frac{1}{3} t^{\frac{1}{3} - 1})$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- t^{\frac{1}{3}} + (8 - t) (\frac{1}{3} t^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 5

Combine $- 1$ and $\frac{3}{3}$ .

$- t^{\frac{1}{3}} + (8 - t) (\frac{1}{3} t^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 6

Combine the numerators over the common denominator.

$- t^{\frac{1}{3}} + (8 - t) (\frac{1}{3} t^{\frac{1 - 1 \cdot 3}{3}})$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

$- t^{\frac{1}{3}} + (8 - t) (\frac{1}{3} t^{\frac{1 - 3}{3}})$

Step 7.2

Subtract $3$ from $1$ .

$- t^{\frac{1}{3}} + (8 - t) (\frac{1}{3} t^{\frac{- 2}{3}})$

Step 8

Move the negative in front of the fraction.

$- t^{\frac{1}{3}} + (8 - t) (\frac{1}{3} t^{- \frac{2}{3}})$

Step 9

Combine $\frac{1}{3}$ and $t^{- \frac{2}{3}}$ .

$- t^{\frac{1}{3}} + (8 - t) \frac{t^{- \frac{2}{3}}}{3}$

Step 10

Move $t^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- t^{\frac{1}{3}} + (8 - t) \frac{1}{3 t^{\frac{2}{3}}}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$- t^{\frac{1}{3}} + 8 \frac{1}{3 t^{\frac{2}{3}}} - t \frac{1}{3 t^{\frac{2}{3}}}$

Step 11.2

Combine terms.

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Step 11.2.1

Combine $8$ and $\frac{1}{3 t^{\frac{2}{3}}}$ .

$- t^{\frac{1}{3}} + \frac{8}{3 t^{\frac{2}{3}}} - t \frac{1}{3 t^{\frac{2}{3}}}$

Step 11.2.2

Combine $\frac{1}{3 t^{\frac{2}{3}}}$ and $t$ .

$- t^{\frac{1}{3}} + \frac{8}{3 t^{\frac{2}{3}}} - \frac{t}{3 t^{\frac{2}{3}}}$

Step 11.2.3

Move $t^{\frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$- t^{\frac{1}{3}} + \frac{8}{3 t^{\frac{2}{3}}} - \frac{t \cdot t^{- \frac{2}{3}}}{3}$

Step 11.2.4

Multiply $t$ by $t^{- \frac{2}{3}}$ by adding the exponents.

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Step 11.2.4.1

Multiply $t$ by $t^{- \frac{2}{3}}$ .

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Step 11.2.4.1.1

Raise $t$ to the power of $1$ .

$- t^{\frac{1}{3}} + \frac{8}{3 t^{\frac{2}{3}}} - \frac{t^{1} t^{- \frac{2}{3}}}{3}$

Step 11.2.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- t^{\frac{1}{3}} + \frac{8}{3 t^{\frac{2}{3}}} - \frac{t^{1 - \frac{2}{3}}}{3}$

Step 11.2.4.2

Write $1$ as a fraction with a common denominator.

$- t^{\frac{1}{3}} + \frac{8}{3 t^{\frac{2}{3}}} - \frac{t^{\frac{3}{3} - \frac{2}{3}}}{3}$

Step 11.2.4.3

Combine the numerators over the common denominator.

$- t^{\frac{1}{3}} + \frac{8}{3 t^{\frac{2}{3}}} - \frac{t^{\frac{3 - 2}{3}}}{3}$

Step 11.2.4.4

Subtract $2$ from $3$ .

$- t^{\frac{1}{3}} + \frac{8}{3 t^{\frac{2}{3}}} - \frac{t^{\frac{1}{3}}}{3}$

Step 11.2.5

To write $- t^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- t^{\frac{1}{3}} \cdot \frac{3}{3} - \frac{t^{\frac{1}{3}}}{3} + \frac{8}{3 t^{\frac{2}{3}}}$

Step 11.2.6

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

$\frac{- t^{\frac{1}{3}} \cdot 3}{3} - \frac{t^{\frac{1}{3}}}{3} + \frac{8}{3 t^{\frac{2}{3}}}$

Step 11.2.7

Combine the numerators over the common denominator.

$\frac{- t^{\frac{1}{3}} \cdot 3 - t^{\frac{1}{3}}}{3} + \frac{8}{3 t^{\frac{2}{3}}}$

Step 11.2.8

Multiply $3$ by $- 1$ .

$\frac{- 3 t^{\frac{1}{3}} - t^{\frac{1}{3}}}{3} + \frac{8}{3 t^{\frac{2}{3}}}$

Step 11.2.9

Subtract $t^{\frac{1}{3}}$ from $- 3 t^{\frac{1}{3}}$ .

$\frac{- 4 t^{\frac{1}{3}}}{3} + \frac{8}{3 t^{\frac{2}{3}}}$

Step 11.2.10

Move the negative in front of the fraction.

$- \frac{4 t^{\frac{1}{3}}}{3} + \frac{8}{3 t^{\frac{2}{3}}}$

Step 11.3

Reorder terms.

$\frac{8}{3 t^{\frac{2}{3}}} - \frac{4 t^{\frac{1}{3}}}{3}$