Calculus Examples

$\sqrt[5]{x} + 4 \sqrt{x^{5}}$

Step 1

By the Sum Rule, the derivative of $\sqrt[5]{x} + 4 \sqrt{x^{5}}$ with respect to $x$ is $\frac{d}{d x} [\sqrt[5]{x}] + \frac{d}{d x} [4 \sqrt{x^{5}}]$ .

$\frac{d}{d x} [\sqrt[5]{x}] + \frac{d}{d x} [4 \sqrt{x^{5}}]$

Step 2

Evaluate $\frac{d}{d x} [\sqrt[5]{x}]$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$\frac{d}{d x} [x^{\frac{1}{5}}] + \frac{d}{d x} [4 \sqrt{x^{5}}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{5}$ .

$\frac{1}{5} x^{\frac{1}{5} - 1} + \frac{d}{d x} [4 \sqrt{x^{5}}]$

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}} + \frac{d}{d x} [4 \sqrt{x^{5}}]$

Step 2.4

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}} + \frac{d}{d x} [4 \sqrt{x^{5}}]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}} + \frac{d}{d x} [4 \sqrt{x^{5}}]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $5$ .

$\frac{1}{5} x^{\frac{1 - 5}{5}} + \frac{d}{d x} [4 \sqrt{x^{5}}]$

Step 2.6.2

Subtract $5$ from $1$ .

$\frac{1}{5} x^{\frac{- 4}{5}} + \frac{d}{d x} [4 \sqrt{x^{5}}]$

Step 2.7

Move the negative in front of the fraction.

$\frac{1}{5} x^{- \frac{4}{5}} + \frac{d}{d x} [4 \sqrt{x^{5}}]$

Step 3

Evaluate $\frac{d}{d x} [4 \sqrt{x^{5}}]$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{5}}$ as $x^{\frac{5}{2}}$ .

$\frac{1}{5} x^{- \frac{4}{5}} + \frac{d}{d x} [4 x^{\frac{5}{2}}]$

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{5}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{5}{2}}]$ .

$\frac{1}{5} x^{- \frac{4}{5}} + 4 \frac{d}{d x} [x^{\frac{5}{2}}]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$\frac{1}{5} x^{- \frac{4}{5}} + 4 (\frac{5}{2} x^{\frac{5}{2} - 1})$

Step 3.4

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

$\frac{1}{5} x^{- \frac{4}{5}} + 4 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}})$

Step 3.5

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

$\frac{1}{5} x^{- \frac{4}{5}} + 4 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.6

Combine the numerators over the common denominator.

$\frac{1}{5} x^{- \frac{4}{5}} + 4 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}})$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

$\frac{1}{5} x^{- \frac{4}{5}} + 4 (\frac{5}{2} x^{\frac{5 - 2}{2}})$

Step 3.7.2

Subtract $2$ from $5$ .

$\frac{1}{5} x^{- \frac{4}{5}} + 4 (\frac{5}{2} x^{\frac{3}{2}})$

Step 3.8

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$\frac{1}{5} x^{- \frac{4}{5}} + 4 \frac{5 x^{\frac{3}{2}}}{2}$

Step 3.9

Combine $4$ and $\frac{5 x^{\frac{3}{2}}}{2}$ .

$\frac{1}{5} x^{- \frac{4}{5}} + \frac{4 (5 x^{\frac{3}{2}})}{2}$

Step 3.10

Multiply $5$ by $4$ .

$\frac{1}{5} x^{- \frac{4}{5}} + \frac{20 x^{\frac{3}{2}}}{2}$

Step 3.11

Factor $2$ out of $20 x^{\frac{3}{2}}$ .

$\frac{1}{5} x^{- \frac{4}{5}} + \frac{2 (10 x^{\frac{3}{2}})}{2}$

Step 3.12

Cancel the common factors.

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Step 3.12.1

Factor $2$ out of $2$ .

$\frac{1}{5} x^{- \frac{4}{5}} + \frac{2 (10 x^{\frac{3}{2}})}{2 (1)}$

Step 3.12.2

Cancel the common factor.

$\frac{1}{5} x^{- \frac{4}{5}} + \frac{2 (10 x^{\frac{3}{2}})}{2 \cdot 1}$

Step 3.12.3

Rewrite the expression.

$\frac{1}{5} x^{- \frac{4}{5}} + \frac{10 x^{\frac{3}{2}}}{1}$

Step 3.12.4

Divide $10 x^{\frac{3}{2}}$ by $1$ .

$\frac{1}{5} x^{- \frac{4}{5}} + 10 x^{\frac{3}{2}}$

Step 4

Simplify.

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Step 4.1

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

$\frac{1}{5} \cdot \frac{1}{x^{\frac{4}{5}}} + 10 x^{\frac{3}{2}}$

Step 4.2

Multiply $\frac{1}{5}$ by $\frac{1}{x^{\frac{4}{5}}}$ .

$\frac{1}{5 x^{\frac{4}{5}}} + 10 x^{\frac{3}{2}}$

Step 4.3

Reorder terms.

$10 x^{\frac{3}{2}} + \frac{1}{5 x^{\frac{4}{5}}}$