Calculus Examples

$f (x) = \sqrt{x^{2} + 27}$ , $x = 3$

Step 1

Find the derivative.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 27}$ as ${(x^{2} + 27)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{2} + 27)}^{\frac{1}{2}}]$

Step 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 $g (x) = x^{2} + 27$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 27$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} + 27]$

Step 1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 27]$

Step 1.2.3

Replace all occurrences of $u$ with $x^{2} + 27$ .

$\frac{1}{2} {(x^{2} + 27)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 27]$

Step 1.3

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

$\frac{1}{2} {(x^{2} + 27)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} + 27]$

Step 1.4

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

$\frac{1}{2} {(x^{2} + 27)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 27]$

Step 1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{2} + 27)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 27]$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{2} + 27)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} + 27]$

Step 1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{2} + 27)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} + 27]$

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{2} + 27)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} + 27]$

Step 1.7.2

Combine $\frac{1}{2}$ and ${(x^{2} + 27)}^{- \frac{1}{2}}$ .

$\frac{{(x^{2} + 27)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} + 27]$

Step 1.7.3

Move ${(x^{2} + 27)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{2} + 27)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} + 27]$

Step 1.8

By the Sum Rule, the derivative of $x^{2} + 27$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [27]$ .

$\frac{1}{2 {(x^{2} + 27)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [27])$

Step 1.9

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

$\frac{1}{2 {(x^{2} + 27)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [27])$

Step 1.10

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x^{2} + 27)}^{\frac{1}{2}}} (2 x + 0)$

Step 1.11

Simplify terms.

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Step 1.11.1

Add $2 x$ and $0$ .

$\frac{1}{2 {(x^{2} + 27)}^{\frac{1}{2}}} (2 x)$

Step 1.11.2

Combine $2$ and $\frac{1}{2 {(x^{2} + 27)}^{\frac{1}{2}}}$ .

$\frac{2}{2 {(x^{2} + 27)}^{\frac{1}{2}}} x$

Step 1.11.3

Combine $\frac{2}{2 {(x^{2} + 27)}^{\frac{1}{2}}}$ and $x$ .

$\frac{2 x}{2 {(x^{2} + 27)}^{\frac{1}{2}}}$

Step 1.11.4

Cancel the common factor.

$\frac{2 x}{2 {(x^{2} + 27)}^{\frac{1}{2}}}$

Step 1.11.5

Rewrite the expression.

$\frac{x}{{(x^{2} + 27)}^{\frac{1}{2}}}$

Step 2

Replace the variable $x$ with $3$ in the expression.

$f' (3) = \frac{3}{{({(3)}^{2} + 27)}^{\frac{1}{2}}}$

Step 3

Simplify the denominator.

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Step 3.1

Raise $3$ to the power of $2$ .

$f' (3) = \frac{3}{{(9 + 27)}^{\frac{1}{2}}}$

Step 3.2

Add $9$ and $27$ .

$f' (3) = \frac{3}{36^{\frac{1}{2}}}$

Step 3.3

Rewrite $36$ as $6^{2}$ .

$f' (3) = \frac{3}{{(6^{2})}^{\frac{1}{2}}}$

Step 3.4

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

$f' (3) = \frac{3}{6^{2 (\frac{1}{2})}}$

Step 3.5

Cancel the common factor of $2$ .

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Step 3.5.1

Cancel the common factor.

$f' (3) = \frac{3}{6^{2 (\frac{1}{2})}}$

Step 3.5.2

Rewrite the expression.

$f' (3) = \frac{3}{6}$

Step 3.6

Evaluate the exponent.

$f' (3) = \frac{3}{6}$

Step 4

Cancel the common factor of $3$ and $6$ .

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Step 4.1

Factor $3$ out of $3$ .

$f' (3) = \frac{3 (1)}{6}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $6$ .

$f' (3) = \frac{3 \cdot 1}{3 \cdot 2}$

Step 4.2.2

Cancel the common factor.

$f' (3) = \frac{3 \cdot 1}{3 \cdot 2}$

Step 4.2.3

Rewrite the expression.

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