Calculus Examples

$y = \frac{8 x + 7}{\sqrt{x}}$

Step 1

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

$y = \frac{8 x + 7}{x^{\frac{1}{2}}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{8 x + 7}{x^{\frac{1}{2}}})$

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 $f (x) = 8 x + 7$ and $g (x) = x^{\frac{1}{2}}$ .

$\frac{x^{\frac{1}{2}} \frac{d}{d x} [8 x + 7] - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{{(x^{\frac{1}{2}})}^{2}}$

Step 4.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1

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

$\frac{x^{\frac{1}{2}} \frac{d}{d x} [8 x + 7] - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x^{\frac{1}{2} \cdot 2}}$

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{x^{\frac{1}{2}} \frac{d}{d x} [8 x + 7] - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x^{\frac{1}{2} \cdot 2}}$

Step 4.2.2.2

Rewrite the expression.

$\frac{x^{\frac{1}{2}} \frac{d}{d x} [8 x + 7] - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x^{1}}$

Step 4.3

Simplify.

$\frac{x^{\frac{1}{2}} \frac{d}{d x} [8 x + 7] - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 4.4

By the Sum Rule, the derivative of $8 x + 7$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [7]$ .

$\frac{x^{\frac{1}{2}} (\frac{d}{d x} [8 x] + \frac{d}{d x} [7]) - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 4.5

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\frac{x^{\frac{1}{2}} (8 \frac{d}{d x} [x] + \frac{d}{d x} [7]) - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 4.6

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

$\frac{x^{\frac{1}{2}} (8 \cdot 1 + \frac{d}{d x} [7]) - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 4.7

Multiply $8$ by $1$ .

$\frac{x^{\frac{1}{2}} (8 + \frac{d}{d x} [7]) - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 4.8

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

$\frac{x^{\frac{1}{2}} (8 + 0) - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 4.9

Simplify the expression.

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Step 4.9.1

Add $8$ and $0$ .

$\frac{x^{\frac{1}{2}} \cdot 8 - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 4.9.2

Move $8$ to the left of $x^{\frac{1}{2}}$ .

$\frac{8 \cdot x^{\frac{1}{2}} - (8 x + 7) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 4.10

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

$\frac{8 x^{\frac{1}{2}} - (8 x + 7) (\frac{1}{2} x^{\frac{1}{2} - 1})}{x}$

Step 4.11

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

$\frac{8 x^{\frac{1}{2}} - (8 x + 7) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})}{x}$

Step 4.12

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

$\frac{8 x^{\frac{1}{2}} - (8 x + 7) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})}{x}$

Step 4.13

Combine the numerators over the common denominator.

$\frac{8 x^{\frac{1}{2}} - (8 x + 7) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})}{x}$

Step 4.14

Simplify the numerator.

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Step 4.14.1

Multiply $- 1$ by $2$ .

$\frac{8 x^{\frac{1}{2}} - (8 x + 7) (\frac{1}{2} x^{\frac{1 - 2}{2}})}{x}$

Step 4.14.2

Subtract $2$ from $1$ .

$\frac{8 x^{\frac{1}{2}} - (8 x + 7) (\frac{1}{2} x^{\frac{- 1}{2}})}{x}$

Step 4.15

Move the negative in front of the fraction.

$\frac{8 x^{\frac{1}{2}} - (8 x + 7) (\frac{1}{2} x^{- \frac{1}{2}})}{x}$

Step 4.16

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

$\frac{8 x^{\frac{1}{2}} - (8 x + 7) \frac{x^{- \frac{1}{2}}}{2}}{x}$

Step 4.17

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

$\frac{8 x^{\frac{1}{2}} - (8 x + 7) \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18

Simplify.

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Step 4.18.1

Apply the distributive property.

$\frac{8 x^{\frac{1}{2}} + (- (8 x) - 1 \cdot 7) \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.2

Apply the distributive property.

$\frac{8 x^{\frac{1}{2}} - (8 x) \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3

Simplify the numerator.

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Step 4.18.3.1

Simplify each term.

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Step 4.18.3.1.1

Multiply $8$ by $- 1$ .

$\frac{8 x^{\frac{1}{2}} - 8 x \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.2

Cancel the common factor of $2$ .

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Step 4.18.3.1.2.1

Factor $2$ out of $- 8 x$ .

$\frac{8 x^{\frac{1}{2}} + 2 (- 4 x) \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.2.2

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

$\frac{8 x^{\frac{1}{2}} + 2 (- 4 x) \frac{1}{2 (x^{\frac{1}{2}})} - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.2.3

Cancel the common factor.

$\frac{8 x^{\frac{1}{2}} + 2 (- 4 x) \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.2.4

Rewrite the expression.

$\frac{8 x^{\frac{1}{2}} - 4 x \frac{1}{x^{\frac{1}{2}}} - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.3

Combine $- 4$ and $\frac{1}{x^{\frac{1}{2}}}$ .

$\frac{8 x^{\frac{1}{2}} + x \frac{- 4}{x^{\frac{1}{2}}} - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.4

Combine $x$ and $\frac{- 4}{x^{\frac{1}{2}}}$ .

$\frac{8 x^{\frac{1}{2}} + \frac{x \cdot - 4}{x^{\frac{1}{2}}} - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.5

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

$\frac{8 x^{\frac{1}{2}} + x \cdot - 4 x^{- \frac{1}{2}} - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.6

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

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Step 4.18.3.1.6.1

Move $x^{- \frac{1}{2}}$ .

$\frac{8 x^{\frac{1}{2}} + x^{- \frac{1}{2}} x \cdot - 4 - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.6.2

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

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Step 4.18.3.1.6.2.1

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

$\frac{8 x^{\frac{1}{2}} + x^{- \frac{1}{2}} x^{1} \cdot - 4 - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.6.2.2

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

$\frac{8 x^{\frac{1}{2}} + x^{- \frac{1}{2} + 1} \cdot - 4 - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.6.3

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

$\frac{8 x^{\frac{1}{2}} + x^{- \frac{1}{2} + \frac{2}{2}} \cdot - 4 - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.6.4

Combine the numerators over the common denominator.

$\frac{8 x^{\frac{1}{2}} + x^{\frac{- 1 + 2}{2}} \cdot - 4 - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.6.5

Add $- 1$ and $2$ .

$\frac{8 x^{\frac{1}{2}} + x^{\frac{1}{2}} \cdot - 4 - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.7

Move $- 4$ to the left of $x^{\frac{1}{2}}$ .

$\frac{8 x^{\frac{1}{2}} - 4 \cdot x^{\frac{1}{2}} - 1 \cdot 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.8

Multiply $- 1$ by $7$ .

$\frac{8 x^{\frac{1}{2}} - 4 x^{\frac{1}{2}} - 7 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.9

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

$\frac{8 x^{\frac{1}{2}} - 4 x^{\frac{1}{2}} + \frac{- 7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.1.10

Move the negative in front of the fraction.

$\frac{8 x^{\frac{1}{2}} - 4 x^{\frac{1}{2}} - \frac{7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.3.2

Subtract $4 x^{\frac{1}{2}}$ from $8 x^{\frac{1}{2}}$ .

$\frac{4 x^{\frac{1}{2}} - \frac{7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4

Simplify the numerator.

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Step 4.18.4.1

To write $4 x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$\frac{4 x^{\frac{1}{2}} \cdot \frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - \frac{7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4.2

Combine $4 x^{\frac{1}{2}}$ and $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$\frac{\frac{4 x^{\frac{1}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}} - \frac{7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4.3

Combine the numerators over the common denominator.

$\frac{\frac{4 x^{\frac{1}{2}} (2 x^{\frac{1}{2}}) - 7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4.4

Simplify the numerator.

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Step 4.18.4.4.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{4 \cdot 2 x^{\frac{1}{2}} x^{\frac{1}{2}} - 7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4.4.2

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

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Step 4.18.4.4.2.1

Move $x^{\frac{1}{2}}$ .

$\frac{\frac{4 \cdot 2 (x^{\frac{1}{2}} x^{\frac{1}{2}}) - 7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4.4.2.2

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

$\frac{\frac{4 \cdot 2 x^{\frac{1}{2} + \frac{1}{2}} - 7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4.4.2.3

Combine the numerators over the common denominator.

$\frac{\frac{4 \cdot 2 x^{\frac{1 + 1}{2}} - 7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4.4.2.4

Add $1$ and $1$ .

$\frac{\frac{4 \cdot 2 x^{\frac{2}{2}} - 7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4.4.2.5

Divide $2$ by $2$ .

$\frac{\frac{4 \cdot 2 x^{1} - 7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4.4.3

Simplify $4 \cdot 2 x^{1}$ .

$\frac{\frac{4 \cdot 2 x - 7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.4.4.4

Multiply $4$ by $2$ .

$\frac{\frac{8 x - 7}{2 x^{\frac{1}{2}}}}{x}$

Step 4.18.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{8 x - 7}{2 x^{\frac{1}{2}}} \cdot \frac{1}{x}$

Step 4.18.6

Multiply $\frac{8 x - 7}{2 x^{\frac{1}{2}}} \cdot \frac{1}{x}$ .

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Step 4.18.6.1

Multiply $\frac{8 x - 7}{2 x^{\frac{1}{2}}}$ by $\frac{1}{x}$ .

$\frac{8 x - 7}{2 x^{\frac{1}{2}} x}$

Step 4.18.6.2

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

$\frac{8 x - 7}{2 (x^{1} x^{\frac{1}{2}})}$

Step 4.18.6.3

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

$\frac{8 x - 7}{2 x^{1 + \frac{1}{2}}}$

Step 4.18.6.4

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

$\frac{8 x - 7}{2 x^{\frac{2}{2} + \frac{1}{2}}}$

Step 4.18.6.5

Combine the numerators over the common denominator.

$\frac{8 x - 7}{2 x^{\frac{2 + 1}{2}}}$

Step 4.18.6.6

Add $2$ and $1$ .

$\frac{8 x - 7}{2 x^{\frac{3}{2}}}$

Step 5

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

$y' = \frac{8 x - 7}{2 x^{\frac{3}{2}}}$

Step 6

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

$\frac{d y}{d x} = \frac{8 x - 7}{2 x^{\frac{3}{2}}}$