Calculus Examples

$\sqrt{8 x}$

Step 1

Rewrite $\frac{d}{d x} [\sqrt{8 x}]$ as $\frac{d}{d x} [2 \sqrt{2 x}]$ .

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Step 1.1

Rewrite $8 x$ as $2^{2} \cdot (2 x)$ .

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Step 1.1.1

Factor $4$ out of $8$ .

$\frac{d}{d x} [\sqrt{4 (2) x}]$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{d}{d x} [\sqrt{2^{2} \cdot 2 x}]$

Step 1.1.3

Add parentheses.

$\frac{d}{d x} [\sqrt{2^{2} \cdot (2 x)}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [2 \sqrt{2 x}]$

Step 2

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

$\frac{d}{d x} [2 {(2 x)}^{\frac{1}{2}}]$

Step 3

Factor $2$ out of $2 x$ .

$\frac{d}{d x} [2 {(2 (x))}^{\frac{1}{2}}]$

Step 4

Apply the product rule to $2 (x)$ .

$\frac{d}{d x} [2 (2^{\frac{1}{2}} x^{\frac{1}{2}})]$

Step 5

Raise $2$ to the power of $1$ .

$\frac{d}{d x} [2^{\frac{1}{2}} \cdot 2^{1} x^{\frac{1}{2}}]$

Step 6

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

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

Step 7

Simplify the expression.

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Step 7.1

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

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

Step 7.2

Combine the numerators over the common denominator.

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

Step 7.3

Add $1$ and $2$ .

$\frac{d}{d x} [2^{\frac{3}{2}} x^{\frac{1}{2}}]$

Step 8

Since $2^{\frac{3}{2}}$ is constant with respect to $x$ , the derivative of $2^{\frac{3}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $2^{\frac{3}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$2^{\frac{3}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 9

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

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

Step 10

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

$2^{\frac{3}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

$2^{\frac{3}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 13

Simplify the numerator.

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Step 13.1

Multiply $- 1$ by $2$ .

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

Step 13.2

Subtract $2$ from $1$ .

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

Step 14

Move the negative in front of the fraction.

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

Step 15

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

$2^{\frac{3}{2}} \frac{x^{- \frac{1}{2}}}{2}$

Step 16

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

$\frac{2^{\frac{3}{2}} x^{- \frac{1}{2}}}{2}$

Step 17

Simplify.

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Step 17.1

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

$\frac{2^{\frac{3}{2}}}{2 x^{\frac{1}{2}}}$

Step 17.2

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

$\frac{2^{\frac{3}{2}} \cdot 2^{- 1}}{x^{\frac{1}{2}}}$

Step 18

Multiply $2^{\frac{3}{2}}$ by $2^{- 1}$ by adding the exponents.

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Step 18.1

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

$\frac{2^{\frac{3}{2} - 1}}{x^{\frac{1}{2}}}$

Step 18.2

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

$\frac{2^{\frac{3}{2} - 1 \cdot \frac{2}{2}}}{x^{\frac{1}{2}}}$

Step 18.3

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

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

Step 18.4

Combine the numerators over the common denominator.

$\frac{2^{\frac{3 - 1 \cdot 2}{2}}}{x^{\frac{1}{2}}}$

Step 18.5

Simplify the numerator.

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Step 18.5.1

Multiply $- 1$ by $2$ .

$\frac{2^{\frac{3 - 2}{2}}}{x^{\frac{1}{2}}}$

Step 18.5.2

Subtract $2$ from $3$ .

$\frac{2^{\frac{1}{2}}}{x^{\frac{1}{2}}}$