Calculus Examples

$f (x) = \sqrt{x^{2} + 8}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 8}$ as ${(x^{2} + 8)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{2} + 8)}^{\frac{1}{2}}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{2} + 8$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 8$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} + 8]$

Step 1.1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 8]$

Step 1.1.2.3

Replace all occurrences of $u$ with $x^{2} + 8$ .

$\frac{1}{2} {(x^{2} + 8)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 8]$

Step 1.1.3

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

$\frac{1}{2} {(x^{2} + 8)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} + 8]$

Step 1.1.4

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

$\frac{1}{2} {(x^{2} + 8)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 8]$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{2} + 8)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 8]$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{2} + 8)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} + 8]$

Step 1.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{2} + 8)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} + 8]$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{2} + 8)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} + 8]$

Step 1.1.7.2

Combine $\frac{1}{2}$ and ${(x^{2} + 8)}^{- \frac{1}{2}}$ .

$\frac{{(x^{2} + 8)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} + 8]$

Step 1.1.7.3

Move ${(x^{2} + 8)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{2} + 8)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} + 8]$

Step 1.1.8

By the Sum Rule, the derivative of $x^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8]$ .

$\frac{1}{2 {(x^{2} + 8)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8])$

Step 1.1.9

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

$\frac{1}{2 {(x^{2} + 8)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [8])$

Step 1.1.10

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x^{2} + 8)}^{\frac{1}{2}}} (2 x + 0)$

Step 1.1.11

Simplify terms.

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Step 1.1.11.1

Add $2 x$ and $0$ .

$\frac{1}{2 {(x^{2} + 8)}^{\frac{1}{2}}} (2 x)$

Step 1.1.11.2

Combine $2$ and $\frac{1}{2 {(x^{2} + 8)}^{\frac{1}{2}}}$ .

$\frac{2}{2 {(x^{2} + 8)}^{\frac{1}{2}}} x$

Step 1.1.11.3

Combine $\frac{2}{2 {(x^{2} + 8)}^{\frac{1}{2}}}$ and $x$ .

$\frac{2 x}{2 {(x^{2} + 8)}^{\frac{1}{2}}}$

Step 1.1.11.4

Cancel the common factor.

$\frac{2 x}{2 {(x^{2} + 8)}^{\frac{1}{2}}}$

Step 1.1.11.5

Rewrite the expression.

$f' (x) = \frac{x}{{(x^{2} + 8)}^{\frac{1}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x}{{(x^{2} + 8)}^{\frac{1}{2}}}$ .

$\frac{x}{{(x^{2} + 8)}^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x}{{(x^{2} + 8)}^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{x}{{(x^{2} + 8)}^{\frac{1}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$x = 0$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\sqrt{x^{2} + 8}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\sqrt{{(0)}^{2} + 8}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Raising $0$ to any positive power yields $0$ .

$\sqrt{0 + 8}$

Step 4.1.2.2

Add $0$ and $8$ .

$\sqrt{8}$

Step 4.1.2.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.1.2.3.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)}$

Step 4.1.2.3.2

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

$\sqrt{2^{2} \cdot 2}$

Step 4.1.2.4

Pull terms out from under the radical.

$2 \sqrt{2}$

Step 4.2

List all of the points.

$(0, 2 \sqrt{2})$

Step 5