Calculus Examples

$\frac{\sqrt{x} - 9}{\sqrt{x} + 9}$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

$\frac{d}{d x} [\frac{x^{\frac{1}{2}} - 9}{\sqrt{x} + 9}]$

Step 1.2

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

$\frac{d}{d x} [\frac{x^{\frac{1}{2}} - 9}{x^{\frac{1}{2}} + 9}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{\frac{1}{2}} - 9$ and $g (x) = x^{\frac{1}{2}} + 9$ .

$\frac{(x^{\frac{1}{2}} + 9) \frac{d}{d x} [x^{\frac{1}{2}} - 9] - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{\frac{1}{2}} - 9$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 9]$ .

$\frac{(x^{\frac{1}{2}} + 9) (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 9]) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 3.2

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

$\frac{(x^{\frac{1}{2}} + 9) (\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [- 9]) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 4

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

$\frac{(x^{\frac{1}{2}} + 9) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- 9]) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 5

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

$\frac{(x^{\frac{1}{2}} + 9) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- 9]) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 6

Combine the numerators over the common denominator.

$\frac{(x^{\frac{1}{2}} + 9) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [- 9]) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$\frac{(x^{\frac{1}{2}} + 9) (\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [- 9]) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 7.2

Subtract $2$ from $1$ .

$\frac{(x^{\frac{1}{2}} + 9) (\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [- 9]) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$\frac{(x^{\frac{1}{2}} + 9) (\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [- 9]) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 8.2

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

$\frac{(x^{\frac{1}{2}} + 9) (\frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 9]) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 8.3

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

$\frac{(x^{\frac{1}{2}} + 9) (\frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 9]) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 9

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{(x^{\frac{1}{2}} + 9) (\frac{1}{2 x^{\frac{1}{2}}} + 0) - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 10

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x^{\frac{1}{2}} + 9]}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 11

By the Sum Rule, the derivative of $x^{\frac{1}{2}} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [9]$ .

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [9])}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 12

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

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [9])}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 13

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

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [9])}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 14

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

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [9])}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 15

Combine the numerators over the common denominator.

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [9])}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 16

Simplify the numerator.

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Step 16.1

Multiply $- 1$ by $2$ .

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [9])}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 16.2

Subtract $2$ from $1$ .

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [9])}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 17

Combine fractions.

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Step 17.1

Move the negative in front of the fraction.

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [9])}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 17.2

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

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [9])}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 17.3

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

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [9])}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 18

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) (\frac{1}{2 x^{\frac{1}{2}}} + 0)}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 19

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$\frac{(x^{\frac{1}{2}} + 9) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20

Simplify.

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Step 20.1

Apply the distributive property.

$\frac{x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} + 9 \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} - 9) \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.2

Apply the distributive property.

$\frac{x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} + 9 \frac{1}{2 x^{\frac{1}{2}}} + (- x^{\frac{1}{2}} - - 9) \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.3

Apply the distributive property.

$\frac{x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} + 9 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} - - 9 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4

Simplify the numerator.

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Step 20.4.1

Combine the opposite terms in $x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} + 9 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} - - 9 \frac{1}{2 x^{\frac{1}{2}}}$ .

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Step 20.4.1.1

Subtract $x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}$ from $x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{9 \frac{1}{2 x^{\frac{1}{2}}} + 0 - - 9 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4.1.2

Add $9 \frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$\frac{9 \frac{1}{2 x^{\frac{1}{2}}} - - 9 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4.2

Simplify each term.

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Step 20.4.2.1

Combine $9$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{\frac{9}{2 x^{\frac{1}{2}}} - - 9 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4.2.2

Multiply $- 1$ by $- 9$ .

$\frac{\frac{9}{2 x^{\frac{1}{2}}} + 9 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4.2.3

Combine $9$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{\frac{9}{2 x^{\frac{1}{2}}} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4.3

Combine the numerators over the common denominator.

$\frac{\frac{9 + 9}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4.4

Add $9$ and $9$ .

$\frac{\frac{18}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4.5

Factor $2$ out of $18$ .

$\frac{\frac{2 \cdot 9}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4.6

Cancel the common factors.

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Step 20.4.6.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$\frac{\frac{2 \cdot 9}{2 (x^{\frac{1}{2}})}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4.6.2

Cancel the common factor.

$\frac{\frac{2 \cdot 9}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.4.6.3

Rewrite the expression.

$\frac{\frac{9}{x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.5

Combine terms.

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Step 20.5.1

Rewrite $\frac{\frac{9}{x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 9)}^{2}}$ as a product.

$\frac{9}{x^{\frac{1}{2}}} \cdot \frac{1}{{(x^{\frac{1}{2}} + 9)}^{2}}$

Step 20.5.2

Multiply $\frac{9}{x^{\frac{1}{2}}}$ by $\frac{1}{{(x^{\frac{1}{2}} + 9)}^{2}}$ .

$\frac{9}{x^{\frac{1}{2}} {(x^{\frac{1}{2}} + 9)}^{2}}$