Calculus Examples

$f (u) = \sqrt{u} (5 - u^{6})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d u} [f (u) g (u)]$ is $f (u) \frac{d}{d u} [g (u)] + g (u) \frac{d}{d u} [f (u)]$ where $f (u) = \sqrt{u}$ and $g (u) = 5 - u^{6}$ .

$\sqrt{u} \frac{d}{d u} [5 - u^{6}] + (5 - u^{6}) \frac{d}{d u} [\sqrt{u}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $5 - u^{6}$ with respect to $u$ is $\frac{d}{d u} [5] + \frac{d}{d u} [- u^{6}]$ .

$\sqrt{u} (\frac{d}{d u} [5] + \frac{d}{d u} [- u^{6}]) + (5 - u^{6}) \frac{d}{d u} [\sqrt{u}]$

Step 2.2

Since $5$ is constant with respect to $u$ , the derivative of $5$ with respect to $u$ is $0$ .

$\sqrt{u} (0 + \frac{d}{d u} [- u^{6}]) + (5 - u^{6}) \frac{d}{d u} [\sqrt{u}]$

Step 3

Evaluate $\frac{d}{d u} [- u^{6}]$ .

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Step 3.1

Since $- 1$ is constant with respect to $u$ , the derivative of $- u^{6}$ with respect to $u$ is $- \frac{d}{d u} [u^{6}]$ .

$\sqrt{u} (0 - \frac{d}{d u} [u^{6}]) + (5 - u^{6}) \frac{d}{d u} [\sqrt{u}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$\sqrt{u} (0 - (6 u^{5})) + (5 - u^{6}) \frac{d}{d u} [\sqrt{u}]$

Step 3.3

Multiply $6$ by $- 1$ .

$\sqrt{u} (0 - 6 u^{5}) + (5 - u^{6}) \frac{d}{d u} [\sqrt{u}]$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Subtract $6 u^{5}$ from $0$ .

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) \frac{d}{d u} [\sqrt{u}]$

Step 4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) \frac{d}{d u} [u^{\frac{1}{2}}]$

Step 4.3

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

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) (\frac{1}{2} u^{\frac{1}{2} - 1})$

Step 5

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

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) (\frac{1}{2} u^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 6

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

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) (\frac{1}{2} u^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 7

Combine the numerators over the common denominator.

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) (\frac{1}{2} u^{\frac{1 - 1 \cdot 2}{2}})$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) (\frac{1}{2} u^{\frac{1 - 2}{2}})$

Step 8.2

Subtract $2$ from $1$ .

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) (\frac{1}{2} u^{\frac{- 1}{2}})$

Step 9

Move the negative in front of the fraction.

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) (\frac{1}{2} u^{- \frac{1}{2}})$

Step 10

Simplify.

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Step 10.1

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

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) (\frac{1}{2} \cdot \frac{1}{u^{\frac{1}{2}}})$

Step 10.2

Multiply $\frac{1}{2}$ by $\frac{1}{u^{\frac{1}{2}}}$ .

$\sqrt{u} (- 6 u^{5}) + (5 - u^{6}) \frac{1}{2 u^{\frac{1}{2}}}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$\sqrt{u} (- 6 u^{5}) + 5 \frac{1}{2 u^{\frac{1}{2}}} - u^{6} \frac{1}{2 u^{\frac{1}{2}}}$

Step 11.2

Combine terms.

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Step 11.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 11.2.2

Multiply $u^{\frac{1}{2}}$ by $u^{5}$ by adding the exponents.

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Step 11.2.2.1

Move $u^{5}$ .

$u^{5} u^{\frac{1}{2}} \cdot - 6 + 5 \frac{1}{2 u^{\frac{1}{2}}} - u^{6} \frac{1}{2 u^{\frac{1}{2}}}$

Step 11.2.2.2

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

$u^{5 + \frac{1}{2}} \cdot - 6 + 5 \frac{1}{2 u^{\frac{1}{2}}} - u^{6} \frac{1}{2 u^{\frac{1}{2}}}$

Step 11.2.2.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$u^{5 \cdot \frac{2}{2} + \frac{1}{2}} \cdot - 6 + 5 \frac{1}{2 u^{\frac{1}{2}}} - u^{6} \frac{1}{2 u^{\frac{1}{2}}}$

Step 11.2.2.4

Combine $5$ and $\frac{2}{2}$ .

$u^{\frac{5 \cdot 2}{2} + \frac{1}{2}} \cdot - 6 + 5 \frac{1}{2 u^{\frac{1}{2}}} - u^{6} \frac{1}{2 u^{\frac{1}{2}}}$

Step 11.2.2.5

Combine the numerators over the common denominator.

$u^{\frac{5 \cdot 2 + 1}{2}} \cdot - 6 + 5 \frac{1}{2 u^{\frac{1}{2}}} - u^{6} \frac{1}{2 u^{\frac{1}{2}}}$

Step 11.2.2.6

Simplify the numerator.

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Step 11.2.2.6.1

Multiply $5$ by $2$ .

$u^{\frac{10 + 1}{2}} \cdot - 6 + 5 \frac{1}{2 u^{\frac{1}{2}}} - u^{6} \frac{1}{2 u^{\frac{1}{2}}}$

Step 11.2.2.6.2

Add $10$ and $1$ .

$u^{\frac{11}{2}} \cdot - 6 + 5 \frac{1}{2 u^{\frac{1}{2}}} - u^{6} \frac{1}{2 u^{\frac{1}{2}}}$

Step 11.2.3

Move $- 6$ to the left of $u^{\frac{11}{2}}$ .

$- 6 \cdot u^{\frac{11}{2}} + 5 \frac{1}{2 u^{\frac{1}{2}}} - u^{6} \frac{1}{2 u^{\frac{1}{2}}}$

Step 11.2.4

Combine $5$ and $\frac{1}{2 u^{\frac{1}{2}}}$ .

$- 6 u^{\frac{11}{2}} + \frac{5}{2 u^{\frac{1}{2}}} - u^{6} \frac{1}{2 u^{\frac{1}{2}}}$

Step 11.2.5

Combine $\frac{1}{2 u^{\frac{1}{2}}}$ and $u^{6}$ .

$- 6 u^{\frac{11}{2}} + \frac{5}{2 u^{\frac{1}{2}}} - \frac{u^{6}}{2 u^{\frac{1}{2}}}$

Step 11.2.6

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

$- 6 u^{\frac{11}{2}} + \frac{5}{2 u^{\frac{1}{2}}} - \frac{u^{6} u^{- \frac{1}{2}}}{2}$

Step 11.2.7

Multiply $u^{6}$ by $u^{- \frac{1}{2}}$ by adding the exponents.

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Step 11.2.7.1

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

$- 6 u^{\frac{11}{2}} + \frac{5}{2 u^{\frac{1}{2}}} - \frac{u^{6 - \frac{1}{2}}}{2}$

Step 11.2.7.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 6 u^{\frac{11}{2}} + \frac{5}{2 u^{\frac{1}{2}}} - \frac{u^{6 \cdot \frac{2}{2} - \frac{1}{2}}}{2}$

Step 11.2.7.3

Combine $6$ and $\frac{2}{2}$ .

$- 6 u^{\frac{11}{2}} + \frac{5}{2 u^{\frac{1}{2}}} - \frac{u^{\frac{6 \cdot 2}{2} - \frac{1}{2}}}{2}$

Step 11.2.7.4

Combine the numerators over the common denominator.

$- 6 u^{\frac{11}{2}} + \frac{5}{2 u^{\frac{1}{2}}} - \frac{u^{\frac{6 \cdot 2 - 1}{2}}}{2}$

Step 11.2.7.5

Simplify the numerator.

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Step 11.2.7.5.1

Multiply $6$ by $2$ .

$- 6 u^{\frac{11}{2}} + \frac{5}{2 u^{\frac{1}{2}}} - \frac{u^{\frac{12 - 1}{2}}}{2}$

Step 11.2.7.5.2

Subtract $1$ from $12$ .

$- 6 u^{\frac{11}{2}} + \frac{5}{2 u^{\frac{1}{2}}} - \frac{u^{\frac{11}{2}}}{2}$

Step 11.2.8

To write $- 6 u^{\frac{11}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 6 u^{\frac{11}{2}} \cdot \frac{2}{2} - \frac{u^{\frac{11}{2}}}{2} + \frac{5}{2 u^{\frac{1}{2}}}$

Step 11.2.9

Combine $- 6 u^{\frac{11}{2}}$ and $\frac{2}{2}$ .

$\frac{- 6 u^{\frac{11}{2}} \cdot 2}{2} - \frac{u^{\frac{11}{2}}}{2} + \frac{5}{2 u^{\frac{1}{2}}}$

Step 11.2.10

Combine the numerators over the common denominator.

$\frac{- 6 u^{\frac{11}{2}} \cdot 2 - u^{\frac{11}{2}}}{2} + \frac{5}{2 u^{\frac{1}{2}}}$

Step 11.2.11

Multiply $2$ by $- 6$ .

$\frac{- 12 u^{\frac{11}{2}} - u^{\frac{11}{2}}}{2} + \frac{5}{2 u^{\frac{1}{2}}}$

Step 11.2.12

Subtract $u^{\frac{11}{2}}$ from $- 12 u^{\frac{11}{2}}$ .

$\frac{- 13 u^{\frac{11}{2}}}{2} + \frac{5}{2 u^{\frac{1}{2}}}$

Step 11.2.13

Move the negative in front of the fraction.

$- \frac{13 u^{\frac{11}{2}}}{2} + \frac{5}{2 u^{\frac{1}{2}}}$

Step 11.3

Reorder terms.

$\frac{5}{2 u^{\frac{1}{2}}} - \frac{13 u^{\frac{11}{2}}}{2}$