Calculus Examples

$f (x) = \sqrt{x^{2} + 21}$ , $a = 2$

Step 1

Consider the function used to find the linearization at $a$ .

$L (x) = f (a) + f' (a) (x - a)$

Step 2

Substitute the value of $a = 2$ into the linearization function.

$L (x) = f (2) + f' (2) (x - 2)$

Step 3

Evaluate $f (2)$ .

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Step 3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \sqrt{{(2)}^{2} + 21}$

Step 3.2

Simplify $\sqrt{{(2)}^{2} + 21}$ .

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Step 3.2.1

Remove parentheses.

$\sqrt{{(2)}^{2} + 21}$

Step 3.2.2

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

$\sqrt{4 + 21}$

Step 3.2.3

Add $4$ and $21$ .

$\sqrt{25}$

Step 3.2.4

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2}}$

Step 3.2.5

Pull terms out from under the radical, assuming positive real numbers.

$5$

Step 4

Find the derivative and evaluate it at $2$ .

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Step 4.1

Find the derivative of $f (x) = \sqrt{x^{2} + 21}$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 21}$ as ${(x^{2} + 21)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{2} + 21)}^{\frac{1}{2}}]$

Step 4.1.2

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 $g (x) = x^{2} + 21$ .

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Step 4.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 21$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} + 21]$

Step 4.1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 21]$

Step 4.1.2.3

Replace all occurrences of $u$ with $x^{2} + 21$ .

$\frac{1}{2} {(x^{2} + 21)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 21]$

Step 4.1.3

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

$\frac{1}{2} {(x^{2} + 21)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} + 21]$

Step 4.1.4

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

$\frac{1}{2} {(x^{2} + 21)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 21]$

Step 4.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{2} + 21)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 21]$

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{2} + 21)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} + 21]$

Step 4.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{2} + 21)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} + 21]$

Step 4.1.7

Combine fractions.

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Step 4.1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{2} + 21)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} + 21]$

Step 4.1.7.2

Combine $\frac{1}{2}$ and ${(x^{2} + 21)}^{- \frac{1}{2}}$ .

$\frac{{(x^{2} + 21)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} + 21]$

Step 4.1.7.3

Move ${(x^{2} + 21)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{2} + 21)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} + 21]$

Step 4.1.8

By the Sum Rule, the derivative of $x^{2} + 21$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [21]$ .

$\frac{1}{2 {(x^{2} + 21)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [21])$

Step 4.1.9

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

$\frac{1}{2 {(x^{2} + 21)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [21])$

Step 4.1.10

Since $21$ is constant with respect to $x$ , the derivative of $21$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x^{2} + 21)}^{\frac{1}{2}}} (2 x + 0)$

Step 4.1.11

Simplify terms.

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Step 4.1.11.1

Add $2 x$ and $0$ .

$\frac{1}{2 {(x^{2} + 21)}^{\frac{1}{2}}} (2 x)$

Step 4.1.11.2

Combine $2$ and $\frac{1}{2 {(x^{2} + 21)}^{\frac{1}{2}}}$ .

$\frac{2}{2 {(x^{2} + 21)}^{\frac{1}{2}}} x$

Step 4.1.11.3

Combine $\frac{2}{2 {(x^{2} + 21)}^{\frac{1}{2}}}$ and $x$ .

$\frac{2 x}{2 {(x^{2} + 21)}^{\frac{1}{2}}}$

Step 4.1.11.4

Cancel the common factor.

$\frac{2 x}{2 {(x^{2} + 21)}^{\frac{1}{2}}}$

Step 4.1.11.5

Rewrite the expression.

$\frac{x}{{(x^{2} + 21)}^{\frac{1}{2}}}$

Step 4.2

Replace the variable $x$ with $2$ in the expression.

$\frac{2}{{({(2)}^{2} + 21)}^{\frac{1}{2}}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

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

$\frac{2}{{(4 + 21)}^{\frac{1}{2}}}$

Step 4.3.2

Add $4$ and $21$ .

$\frac{2}{25^{\frac{1}{2}}}$

Step 4.3.3

Rewrite $25$ as $5^{2}$ .

$\frac{2}{{(5^{2})}^{\frac{1}{2}}}$

Step 4.3.4

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

$\frac{2}{5^{2 (\frac{1}{2})}}$

Step 4.3.5

Cancel the common factor of $2$ .

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Step 4.3.5.1

Cancel the common factor.

$\frac{2}{5^{2 (\frac{1}{2})}}$

Step 4.3.5.2

Rewrite the expression.

$\frac{2}{5^{1}}$

Step 4.3.6

Evaluate the exponent.

$\frac{2}{5}$

Step 5

Substitute the components into the linearization function in order to find the linearization at $a$ .

$L (x) = 5 + \frac{2}{5} (x - 2)$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

$L (x) = 5 + \frac{2}{5} x + \frac{2}{5} \cdot - 2$

Step 6.1.2

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

$L (x) = 5 + \frac{2 x}{5} + \frac{2}{5} \cdot - 2$

Step 6.1.3

Multiply $\frac{2}{5} \cdot - 2$ .

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Step 6.1.3.1

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

$L (x) = 5 + \frac{2 x}{5} + \frac{2 \cdot - 2}{5}$

Step 6.1.3.2

Multiply $2$ by $- 2$ .

$L (x) = 5 + \frac{2 x}{5} + \frac{- 4}{5}$

Step 6.1.4

Move the negative in front of the fraction.

$L (x) = 5 + \frac{2 x}{5} - \frac{4}{5}$

Step 6.2

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

$L (x) = \frac{2 x}{5} + 5 \cdot \frac{5}{5} - \frac{4}{5}$

Step 6.3

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

$L (x) = \frac{2 x}{5} + \frac{5 \cdot 5}{5} - \frac{4}{5}$

Step 6.4

Combine the numerators over the common denominator.

$L (x) = \frac{2 x}{5} + \frac{5 \cdot 5 - 4}{5}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $5$ by $5$ .

$L (x) = \frac{2 x}{5} + \frac{25 - 4}{5}$

Step 6.5.2

Subtract $4$ from $25$ .

$L (x) = \frac{2 x}{5} + \frac{21}{5}$

Step 7