Calculus Examples

$f (x) = \sqrt{8 x}$

Step 1

Find the first derivative.

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Step 1.1

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

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Step 1.1.1

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

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Step 1.1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.1.2

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

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

Step 1.1.1.3

Add parentheses.

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

Step 1.1.2

Pull terms out from under the radical.

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

Step 1.2

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 1.3

Factor $2$ out of $2 x$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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 1.7

Simplify the expression.

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Step 1.7.1

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 1.7.2

Combine the numerators over the common denominator.

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

Step 1.7.3

Add $1$ and $2$ .

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

Step 1.8

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 1.9

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 1.10

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 1.11

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

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

Step 1.12

Combine the numerators over the common denominator.

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

Step 1.13

Simplify the numerator.

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Step 1.13.1

Multiply $- 1$ by $2$ .

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

Step 1.13.2

Subtract $2$ from $1$ .

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

Step 1.14

Move the negative in front of the fraction.

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

Step 1.15

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

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

Step 1.16

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

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

Step 1.17

Simplify.

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Step 1.17.1

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 1.17.2

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 1.18

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

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Step 1.18.1

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 1.18.2

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 1.18.3

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

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

Step 1.18.4

Combine the numerators over the common denominator.

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

Step 1.18.5

Simplify the numerator.

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Step 1.18.5.1

Multiply $- 1$ by $2$ .

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

Step 1.18.5.2

Subtract $2$ from $3$ .

$f' (x) = \frac{2^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

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

Step 2.2.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 2.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

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

Step 2.7.2

Subtract $2$ from $- 1$ .

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

Step 2.8

Move the negative in front of the fraction.

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Simplify the expression.

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Step 2.11.1

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

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

Step 2.11.2

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

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

Step 2.12

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

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Step 2.12.1

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

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Step 2.12.1.1

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

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

Step 2.12.1.2

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

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

Step 2.12.2

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

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

Step 2.12.3

Combine the numerators over the common denominator.

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

Step 2.12.4

Subtract $1$ from $2$ .

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

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{1}{2^{\frac{1}{2}} x^{\frac{3}{2}}}$ .

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