Calculus Examples

$u = \sqrt[7]{t} + 6 \sqrt{t^{7}}$

Step 1

Rewrite the right side with rational exponents.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{t}$ as $t^{\frac{1}{7}}$ .

$u = t^{\frac{1}{7}} + 6 \sqrt{t^{7}}$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t^{7}}$ as $t^{\frac{7}{2}}$ .

$u = t^{\frac{1}{7}} + 6 t^{\frac{7}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d t} (u) = \frac{d}{d t} (t^{\frac{1}{7}} + 6 t^{\frac{7}{2}})$

Step 3

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

$u'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

By the Sum Rule, the derivative of $t^{\frac{1}{7}} + 6 t^{\frac{7}{2}}$ with respect to $t$ is $\frac{d}{d t} [t^{\frac{1}{7}}] + \frac{d}{d t} [6 t^{\frac{7}{2}}]$ .

$\frac{d}{d t} [t^{\frac{1}{7}}] + \frac{d}{d t} [6 t^{\frac{7}{2}}]$

Step 4.2

Evaluate $\frac{d}{d t} [t^{\frac{1}{7}}]$ .

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Step 4.2.1

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{7}$ .

$\frac{1}{7} t^{\frac{1}{7} - 1} + \frac{d}{d t} [6 t^{\frac{7}{2}}]$

Step 4.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{1}{7} t^{\frac{1}{7} - 1 \cdot \frac{7}{7}} + \frac{d}{d t} [6 t^{\frac{7}{2}}]$

Step 4.2.3

Combine $- 1$ and $\frac{7}{7}$ .

$\frac{1}{7} t^{\frac{1}{7} + \frac{- 1 \cdot 7}{7}} + \frac{d}{d t} [6 t^{\frac{7}{2}}]$

Step 4.2.4

Combine the numerators over the common denominator.

$\frac{1}{7} t^{\frac{1 - 1 \cdot 7}{7}} + \frac{d}{d t} [6 t^{\frac{7}{2}}]$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 1$ by $7$ .

$\frac{1}{7} t^{\frac{1 - 7}{7}} + \frac{d}{d t} [6 t^{\frac{7}{2}}]$

Step 4.2.5.2

Subtract $7$ from $1$ .

$\frac{1}{7} t^{\frac{- 6}{7}} + \frac{d}{d t} [6 t^{\frac{7}{2}}]$

Step 4.2.6

Move the negative in front of the fraction.

$\frac{1}{7} t^{- \frac{6}{7}} + \frac{d}{d t} [6 t^{\frac{7}{2}}]$

Step 4.3

Evaluate $\frac{d}{d t} [6 t^{\frac{7}{2}}]$ .

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Step 4.3.1

Since $6$ is constant with respect to $t$ , the derivative of $6 t^{\frac{7}{2}}$ with respect to $t$ is $6 \frac{d}{d t} [t^{\frac{7}{2}}]$ .

$\frac{1}{7} t^{- \frac{6}{7}} + 6 \frac{d}{d t} [t^{\frac{7}{2}}]$

Step 4.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{7}{2}$ .

$\frac{1}{7} t^{- \frac{6}{7}} + 6 (\frac{7}{2} t^{\frac{7}{2} - 1})$

Step 4.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{7} t^{- \frac{6}{7}} + 6 (\frac{7}{2} t^{\frac{7}{2} - 1 \cdot \frac{2}{2}})$

Step 4.3.4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{7} t^{- \frac{6}{7}} + 6 (\frac{7}{2} t^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}})$

Step 4.3.5

Combine the numerators over the common denominator.

$\frac{1}{7} t^{- \frac{6}{7}} + 6 (\frac{7}{2} t^{\frac{7 - 1 \cdot 2}{2}})$

Step 4.3.6

Simplify the numerator.

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Step 4.3.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{7} t^{- \frac{6}{7}} + 6 (\frac{7}{2} t^{\frac{7 - 2}{2}})$

Step 4.3.6.2

Subtract $2$ from $7$ .

$\frac{1}{7} t^{- \frac{6}{7}} + 6 (\frac{7}{2} t^{\frac{5}{2}})$

Step 4.3.7

Combine $\frac{7}{2}$ and $t^{\frac{5}{2}}$ .

$\frac{1}{7} t^{- \frac{6}{7}} + 6 \frac{7 t^{\frac{5}{2}}}{2}$

Step 4.3.8

Combine $6$ and $\frac{7 t^{\frac{5}{2}}}{2}$ .

$\frac{1}{7} t^{- \frac{6}{7}} + \frac{6 (7 t^{\frac{5}{2}})}{2}$

Step 4.3.9

Multiply $7$ by $6$ .

$\frac{1}{7} t^{- \frac{6}{7}} + \frac{42 t^{\frac{5}{2}}}{2}$

Step 4.3.10

Factor $2$ out of $42 t^{\frac{5}{2}}$ .

$\frac{1}{7} t^{- \frac{6}{7}} + \frac{2 (21 t^{\frac{5}{2}})}{2}$

Step 4.3.11

Cancel the common factors.

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Step 4.3.11.1

Factor $2$ out of $2$ .

$\frac{1}{7} t^{- \frac{6}{7}} + \frac{2 (21 t^{\frac{5}{2}})}{2 (1)}$

Step 4.3.11.2

Cancel the common factor.

$\frac{1}{7} t^{- \frac{6}{7}} + \frac{2 (21 t^{\frac{5}{2}})}{2 \cdot 1}$

Step 4.3.11.3

Rewrite the expression.

$\frac{1}{7} t^{- \frac{6}{7}} + \frac{21 t^{\frac{5}{2}}}{1}$

Step 4.3.11.4

Divide $21 t^{\frac{5}{2}}$ by $1$ .

$\frac{1}{7} t^{- \frac{6}{7}} + 21 t^{\frac{5}{2}}$

Step 4.4

Simplify.

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Step 4.4.1

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

$\frac{1}{7} \cdot \frac{1}{t^{\frac{6}{7}}} + 21 t^{\frac{5}{2}}$

Step 4.4.2

Multiply $\frac{1}{7}$ by $\frac{1}{t^{\frac{6}{7}}}$ .

$\frac{1}{7 t^{\frac{6}{7}}} + 21 t^{\frac{5}{2}}$

Step 4.4.3

Reorder terms.

$21 t^{\frac{5}{2}} + \frac{1}{7 t^{\frac{6}{7}}}$

Step 5

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

$u' = 21 t^{\frac{5}{2}} + \frac{1}{7 t^{\frac{6}{7}}}$

Step 6

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

$\frac{d u}{d t} = 21 t^{\frac{5}{2}} + \frac{1}{7 t^{\frac{6}{7}}}$