Calculus Examples

$y = \frac{x - 7}{\sqrt{x} - \sqrt{7}}$

Step 1

Rewrite the right side with rational exponents.

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Step 1.1

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

$y = \frac{x - 7}{x^{\frac{1}{2}} - \sqrt{7}}$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = \frac{x - 7}{x^{\frac{1}{2}} - 7^{\frac{1}{2}}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x - 7}{x^{\frac{1}{2}} - 7^{\frac{1}{2}}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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 $f (x) = x - 7$ and $g (x) = x^{\frac{1}{2}} - 7^{\frac{1}{2}}$ .

$\frac{(x^{\frac{1}{2}} - 7^{\frac{1}{2}}) \frac{d}{d x} [x - 7] - (x - 7) \frac{d}{d x} [x^{\frac{1}{2}} - 7^{\frac{1}{2}}]}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.2

Differentiate.

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Step 4.2.1

By the Sum Rule, the derivative of $x - 7$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 7]$ .

$\frac{(x^{\frac{1}{2}} - 7^{\frac{1}{2}}) (\frac{d}{d x} [x] + \frac{d}{d x} [- 7]) - (x - 7) \frac{d}{d x} [x^{\frac{1}{2}} - 7^{\frac{1}{2}}]}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

$\frac{(x^{\frac{1}{2}} - 7^{\frac{1}{2}}) (1 + \frac{d}{d x} [- 7]) - (x - 7) \frac{d}{d x} [x^{\frac{1}{2}} - 7^{\frac{1}{2}}]}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.2.3

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$\frac{(x^{\frac{1}{2}} - 7^{\frac{1}{2}}) (1 + 0) - (x - 7) \frac{d}{d x} [x^{\frac{1}{2}} - 7^{\frac{1}{2}}]}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.2.4

Simplify the expression.

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Step 4.2.4.1

Add $1$ and $0$ .

$\frac{(x^{\frac{1}{2}} - 7^{\frac{1}{2}}) \cdot 1 - (x - 7) \frac{d}{d x} [x^{\frac{1}{2}} - 7^{\frac{1}{2}}]}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.2.4.2

Multiply $x^{\frac{1}{2}} - 7^{\frac{1}{2}}$ by $1$ .

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) \frac{d}{d x} [x^{\frac{1}{2}} - 7^{\frac{1}{2}}]}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.2.5

By the Sum Rule, the derivative of $x^{\frac{1}{2}} - 7^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 7^{\frac{1}{2}}]$ .

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 7^{\frac{1}{2}}])}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.2.6

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [- 7^{\frac{1}{2}}])}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.3

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- 7^{\frac{1}{2}}])}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.4

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- 7^{\frac{1}{2}}])}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [- 7^{\frac{1}{2}}])}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [- 7^{\frac{1}{2}}])}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.6.2

Subtract $2$ from $1$ .

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [- 7^{\frac{1}{2}}])}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [- 7^{\frac{1}{2}}])}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.7.2

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 7^{\frac{1}{2}}])}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.7.3

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 7^{\frac{1}{2}}])}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.8

Since $- 7^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $- 7^{\frac{1}{2}}$ with respect to $x$ is $0$ .

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) (\frac{1}{2 x^{\frac{1}{2}}} + 0)}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.9

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - (x - 7) \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10

Simplify.

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Step 4.10.1

Apply the distributive property.

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} + (- x - - 7) \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.2

Apply the distributive property.

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - x \frac{1}{2 x^{\frac{1}{2}}} - - 7 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3

Simplify the numerator.

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Step 4.10.3.1

Simplify each term.

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Step 4.10.3.1.1

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - \frac{x}{2 x^{\frac{1}{2}}} - - 7 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.1.2

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - \frac{x \cdot x^{- \frac{1}{2}}}{2} - - 7 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.1.3

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

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Step 4.10.3.1.3.1

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

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Step 4.10.3.1.3.1.1

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - \frac{x^{1} x^{- \frac{1}{2}}}{2} - - 7 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.1.3.1.2

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - \frac{x^{1 - \frac{1}{2}}}{2} - - 7 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.1.3.2

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

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - \frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} - - 7 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.1.3.3

Combine the numerators over the common denominator.

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - \frac{x^{\frac{2 - 1}{2}}}{2} - - 7 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.1.3.4

Subtract $1$ from $2$ .

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} - - 7 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.1.4

Multiply $- 1$ by $- 7$ .

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} + 7 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.1.5

Combine $7$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{x^{\frac{1}{2}} - 7^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} + \frac{7}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.2

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

$\frac{x^{\frac{1}{2}} \cdot \frac{2}{2} - \frac{x^{\frac{1}{2}}}{2} - 7^{\frac{1}{2}} + \frac{7}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.3

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

$\frac{\frac{x^{\frac{1}{2}} \cdot 2}{2} - \frac{x^{\frac{1}{2}}}{2} - 7^{\frac{1}{2}} + \frac{7}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.4

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{1}{2}} \cdot 2 - x^{\frac{1}{2}}}{2} - 7^{\frac{1}{2}} + \frac{7}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.5

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

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Step 4.10.3.5.1

Reorder $x^{\frac{1}{2}}$ and $2$ .

$\frac{\frac{2 \cdot x^{\frac{1}{2}} - x^{\frac{1}{2}}}{2} - 7^{\frac{1}{2}} + \frac{7}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.3.5.2

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

$\frac{\frac{x^{\frac{1}{2}}}{2} - 7^{\frac{1}{2}} + \frac{7}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4

Combine terms.

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Step 4.10.4.1

Multiply $\frac{\frac{x^{\frac{1}{2}}}{2} - 7^{\frac{1}{2}} + \frac{7}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$ by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} \cdot \frac{\frac{x^{\frac{1}{2}}}{2} - 7^{\frac{1}{2}} + \frac{7}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.2

Combine.

$\frac{2 x^{\frac{1}{2}} (\frac{x^{\frac{1}{2}}}{2} - 7^{\frac{1}{2}} + \frac{7}{2 x^{\frac{1}{2}}})}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.3

Apply the distributive property.

$\frac{2 x^{\frac{1}{2}} \frac{x^{\frac{1}{2}}}{2} + 2 x^{\frac{1}{2}} (- 7^{\frac{1}{2}}) + 2 x^{\frac{1}{2}} \frac{7}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.4

Cancel the common factor of $2$ .

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Step 4.10.4.4.1

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

$\frac{2 (x^{\frac{1}{2}}) \frac{x^{\frac{1}{2}}}{2} + 2 x^{\frac{1}{2}} (- 7^{\frac{1}{2}}) + 2 x^{\frac{1}{2}} \frac{7}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.4.2

Cancel the common factor.

Step 4.10.4.4.3

Rewrite the expression.

$\frac{x^{\frac{1}{2}} x^{\frac{1}{2}} + 2 x^{\frac{1}{2}} (- 7^{\frac{1}{2}}) + 2 x^{\frac{1}{2}} \frac{7}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.5

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

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Step 4.10.4.5.1

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

$\frac{x^{\frac{1}{2} + \frac{1}{2}} + 2 x^{\frac{1}{2}} (- 7^{\frac{1}{2}}) + 2 x^{\frac{1}{2}} \frac{7}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.5.2

Combine the numerators over the common denominator.

$\frac{x^{\frac{1 + 1}{2}} + 2 x^{\frac{1}{2}} (- 7^{\frac{1}{2}}) + 2 x^{\frac{1}{2}} \frac{7}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.5.3

Add $1$ and $1$ .

$\frac{x^{\frac{2}{2}} + 2 x^{\frac{1}{2}} (- 7^{\frac{1}{2}}) + 2 x^{\frac{1}{2}} \frac{7}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.5.4

Divide $2$ by $2$ .

$\frac{x^{1} + 2 x^{\frac{1}{2}} (- 7^{\frac{1}{2}}) + 2 x^{\frac{1}{2}} \frac{7}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.6

Simplify $x^{1}$ .

$\frac{x + 2 x^{\frac{1}{2}} (- 7^{\frac{1}{2}}) + 2 x^{\frac{1}{2}} \frac{7}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.7

Cancel the common factor of $2 x^{\frac{1}{2}}$ .

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Step 4.10.4.7.1

Cancel the common factor.

$\frac{x + 2 x^{\frac{1}{2}} (- 7^{\frac{1}{2}}) + 2 x^{\frac{1}{2}} \frac{7}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.7.2

Rewrite the expression.

$\frac{x + 2 x^{\frac{1}{2}} (- 7^{\frac{1}{2}}) + 7}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.4.8

Multiply $- 1$ by $2$ .

$\frac{x - 2 x^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} + 7}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.5

Reorder terms.

$\frac{- 2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}} + x + 7}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.6

Factor $- 1$ out of $- 2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$\frac{- (2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}}) + x + 7}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.7

Factor $- 1$ out of $x$ .

$\frac{- (2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}}) - 1 (- x) + 7}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.8

Factor $- 1$ out of $- (2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}}) - 1 (- x)$ .

$\frac{- (2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}} - x) + 7}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.9

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{- (2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}} - x) - 1 (- 7)}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.10

Factor $- 1$ out of $- (2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}} - x) - 1 (- 7)$ .

$\frac{- (2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}} - x - 7)}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.11

Rewrite $- (2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}} - x - 7)$ as $- 1 (2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}} - x - 7)$ .

$\frac{- 1 (2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}} - x - 7)}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 4.10.12

Move the negative in front of the fraction.

$- \frac{2 \cdot 7^{\frac{1}{2}} x^{\frac{1}{2}} - x - 7}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{2 \cdot (7^{\frac{1}{2}} x^{\frac{1}{2}}) - x - 7}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{2 \cdot (7^{\frac{1}{2}} x^{\frac{1}{2}}) - x - 7}{2 x^{\frac{1}{2}} {(x^{\frac{1}{2}} - 7^{\frac{1}{2}})}^{2}}$