Calculus Examples

$y = \frac{2 x + 5}{\sqrt{x}}$

Step 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 $f (x) = 2 x + 5$ and $g (x) = \sqrt{x}$ .

$\frac{\sqrt{x} \frac{d}{d x} [2 x + 5] - (2 x + 5) \frac{d}{d x} [\sqrt{x}]}{{\sqrt{x}}^{2}}$

Step 2

By the Sum Rule, the derivative of $2 x + 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

$\frac{\sqrt{x} (\frac{d}{d x} [2 x] + \frac{d}{d x} [5]) - (2 x + 5) \frac{d}{d x} [\sqrt{x}]}{{\sqrt{x}}^{2}}$

Step 3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\frac{\sqrt{x} (2 \frac{d}{d x} [x] + \frac{d}{d x} [5]) - (2 x + 5) \frac{d}{d x} [\sqrt{x}]}{{\sqrt{x}}^{2}}$

Step 3.2

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

$\frac{\sqrt{x} (2 \cdot 1 + \frac{d}{d x} [5]) - (2 x + 5) \frac{d}{d x} [\sqrt{x}]}{{\sqrt{x}}^{2}}$

Step 3.3

Multiply $2$ by $1$ .

$\frac{\sqrt{x} (2 + \frac{d}{d x} [5]) - (2 x + 5) \frac{d}{d x} [\sqrt{x}]}{{\sqrt{x}}^{2}}$

Step 4

Differentiate.

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Step 4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{\sqrt{x} (2 + 0) - (2 x + 5) \frac{d}{d x} [\sqrt{x}]}{{\sqrt{x}}^{2}}$

Step 4.2

Simplify the expression.

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Step 4.2.1

Add $2$ and $0$ .

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) \frac{d}{d x} [\sqrt{x}]}{{\sqrt{x}}^{2}}$

Step 4.2.2

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

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) \frac{d}{d x} [x^{\frac{1}{2}}]}{{\sqrt{x}}^{2}}$

Step 4.3

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

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) (\frac{1}{2} x^{\frac{1}{2} - 1})}{{\sqrt{x}}^{2}}$

Step 5

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

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})}{{\sqrt{x}}^{2}}$

Step 6

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

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})}{{\sqrt{x}}^{2}}$

Step 7

Combine the numerators over the common denominator.

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})}{{\sqrt{x}}^{2}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) (\frac{1}{2} x^{\frac{1 - 2}{2}})}{{\sqrt{x}}^{2}}$

Step 8.2

Subtract $2$ from $1$ .

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) (\frac{1}{2} x^{\frac{- 1}{2}})}{{\sqrt{x}}^{2}}$

Step 9

Move the negative in front of the fraction.

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) (\frac{1}{2} x^{- \frac{1}{2}})}{{\sqrt{x}}^{2}}$

Step 10

Simplify.

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Step 10.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) (\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}})}{{\sqrt{x}}^{2}}$

Step 10.2

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

$\frac{\sqrt{x} \cdot 2 - (2 x + 5) \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$\frac{\sqrt{x} \cdot 2 + (- (2 x) - 1 \cdot 5) \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.2

Apply the distributive property.

$\frac{\sqrt{x} \cdot 2 - (2 x) \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3

Simplify each term.

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Step 11.3.1

Move $2$ to the left of $\sqrt{x}$ .

$\frac{2 \cdot \sqrt{x} - (2 x) \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.2

Cancel the common factor of $2$ .

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Step 11.3.2.1

Factor $2$ out of $- (2 x)$ .

$\frac{2 \sqrt{x} + 2 (- (x)) \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.2.2

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

$\frac{2 \sqrt{x} + 2 (- (x)) \frac{1}{2 (x^{\frac{1}{2}})} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.2.3

Cancel the common factor.

$\frac{2 \sqrt{x} + 2 (- (x)) \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.2.4

Rewrite the expression.

$\frac{2 \sqrt{x} - (x) \frac{1}{x^{\frac{1}{2}}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.3

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

$\frac{2 \sqrt{x} - \frac{x}{x^{\frac{1}{2}}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.4

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

$\frac{2 \sqrt{x} - (x \cdot x^{- \frac{1}{2}}) - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.5

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

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Step 11.3.5.1

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

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Step 11.3.5.1.1

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

$\frac{2 \sqrt{x} - (x^{1} x^{- \frac{1}{2}}) - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.5.1.2

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

$\frac{2 \sqrt{x} - x^{1 - \frac{1}{2}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.5.2

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

$\frac{2 \sqrt{x} - x^{\frac{2}{2} - \frac{1}{2}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.5.3

Combine the numerators over the common denominator.

$\frac{2 \sqrt{x} - x^{\frac{2 - 1}{2}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.5.4

Subtract $1$ from $2$ .

$\frac{2 \sqrt{x} - x^{\frac{1}{2}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.6

Multiply $- 1$ by $5$ .

$\frac{2 \sqrt{x} - x^{\frac{1}{2}} - 5 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.7

Combine $- 5$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{2 \sqrt{x} - x^{\frac{1}{2}} + \frac{- 5}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.3.8

Move the negative in front of the fraction.

$\frac{2 \sqrt{x} - x^{\frac{1}{2}} - \frac{5}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 11.4

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 11.4.1

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

$\frac{2 \sqrt{x} - x^{\frac{1}{2}} - \frac{5}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}})}^{2}}$

Step 11.4.2

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

$\frac{2 \sqrt{x} - x^{\frac{1}{2}} - \frac{5}{2 x^{\frac{1}{2}}}}{x^{\frac{1}{2} \cdot 2}}$

Step 11.4.3

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

$\frac{2 \sqrt{x} - x^{\frac{1}{2}} - \frac{5}{2 x^{\frac{1}{2}}}}{x^{\frac{2}{2}}}$

Step 11.4.4

Cancel the common factor of $2$ .

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Step 11.4.4.1

Cancel the common factor.

$\frac{2 \sqrt{x} - x^{\frac{1}{2}} - \frac{5}{2 x^{\frac{1}{2}}}}{x^{\frac{2}{2}}}$

Step 11.4.4.2

Rewrite the expression.

$\frac{2 \sqrt{x} - x^{\frac{1}{2}} - \frac{5}{2 x^{\frac{1}{2}}}}{x^{1}}$

Step 11.4.5

Simplify.

$\frac{2 \sqrt{x} - x^{\frac{1}{2}} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 11.5

Reorder terms.

$\frac{- x^{\frac{1}{2}} - \frac{5}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6

Simplify the numerator.

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Step 11.6.1

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

$\frac{- x^{\frac{1}{2}} \cdot \frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - \frac{5}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.2

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

$\frac{\frac{- x^{\frac{1}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}} - \frac{5}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.3

Combine the numerators over the common denominator.

$\frac{\frac{- x^{\frac{1}{2}} (2 x^{\frac{1}{2}}) - 5}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4

Simplify each term.

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Step 11.6.4.1

Simplify the numerator.

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Step 11.6.4.1.1

Factor $- 1$ out of $- x^{\frac{1}{2}} (2 x^{\frac{1}{2}}) - 5$ .

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Step 11.6.4.1.1.1

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

$\frac{\frac{- 1 \cdot 2 x^{\frac{1}{2}} x^{\frac{1}{2}} - 5}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4.1.1.2

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

$\frac{\frac{- (2 x^{\frac{1}{2}} x^{\frac{1}{2}}) - 5}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4.1.1.3

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{\frac{- (2 x^{\frac{1}{2}} x^{\frac{1}{2}}) - 1 (5)}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4.1.1.4

Factor $- 1$ out of $- (2 x^{\frac{1}{2}} x^{\frac{1}{2}}) - 1 (5)$ .

$\frac{\frac{- (2 x^{\frac{1}{2}} x^{\frac{1}{2}} + 5)}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4.1.2

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

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Step 11.6.4.1.2.1

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

$\frac{\frac{- (2 (x^{\frac{1}{2}} x^{\frac{1}{2}}) + 5)}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4.1.2.2

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

$\frac{\frac{- (2 x^{\frac{1}{2} + \frac{1}{2}} + 5)}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4.1.2.3

Combine the numerators over the common denominator.

$\frac{\frac{- (2 x^{\frac{1 + 1}{2}} + 5)}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4.1.2.4

Add $1$ and $1$ .

$\frac{\frac{- (2 x^{\frac{2}{2}} + 5)}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4.1.2.5

Divide $2$ by $2$ .

$\frac{\frac{- (2 x^{1} + 5)}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4.1.3

Simplify $2 x^{1}$ .

$\frac{\frac{- (2 x + 5)}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.4.2

Move the negative in front of the fraction.

$\frac{- \frac{2 x + 5}{2 x^{\frac{1}{2}}} + 2 \sqrt{x}}{x}$

Step 11.6.5

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

$\frac{- \frac{2 x + 5}{2 x^{\frac{1}{2}}} + 2 \sqrt{x} \cdot \frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.6

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

$\frac{- \frac{2 x + 5}{2 x^{\frac{1}{2}}} + \frac{2 \sqrt{x} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.7

Combine the numerators over the common denominator.

$\frac{\frac{- (2 x + 5) + 2 \sqrt{x} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8

Simplify the numerator.

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Step 11.6.8.1

Apply the distributive property.

$\frac{\frac{- (2 x) - 1 \cdot 5 + 2 \sqrt{x} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.2

Multiply $2$ by $- 1$ .

$\frac{\frac{- 2 x - 1 \cdot 5 + 2 \sqrt{x} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.3

Multiply $- 1$ by $5$ .

$\frac{\frac{- 2 x - 5 + 2 \sqrt{x} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.4

Multiply $2 \sqrt{x} (2 x^{\frac{1}{2}})$ .

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Step 11.6.8.4.1

Multiply $2$ by $2$ .

$\frac{\frac{- 2 x - 5 + 4 \sqrt{x} x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.4.2

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

$\frac{\frac{- 2 x - 5 + 4 (x^{\frac{1}{2}} x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.4.3

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

$\frac{\frac{- 2 x - 5 + 4 x^{\frac{1}{2} + \frac{1}{2}}}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.4.4

Combine the numerators over the common denominator.

$\frac{\frac{- 2 x - 5 + 4 x^{\frac{1 + 1}{2}}}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.4.5

Add $1$ and $1$ .

$\frac{\frac{- 2 x - 5 + 4 x^{\frac{2}{2}}}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.4.6

Cancel the common factor of $2$ .

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Step 11.6.8.4.6.1

Cancel the common factor.

$\frac{\frac{- 2 x - 5 + 4 x^{\frac{2}{2}}}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.4.6.2

Rewrite the expression.

$\frac{\frac{- 2 x - 5 + 4 x^{1}}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.5

Simplify.

$\frac{\frac{- 2 x - 5 + 4 x}{2 x^{\frac{1}{2}}}}{x}$

Step 11.6.8.6

Add $- 2 x$ and $4 x$ .

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

Step 11.7

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x - 5}{2 x^{\frac{1}{2}}} \cdot \frac{1}{x}$

Step 11.8

Multiply $\frac{2 x - 5}{2 x^{\frac{1}{2}}} \cdot \frac{1}{x}$ .

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Step 11.8.1

Multiply $\frac{2 x - 5}{2 x^{\frac{1}{2}}}$ by $\frac{1}{x}$ .

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

Step 11.8.2

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

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

Step 11.8.3

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

$\frac{2 x - 5}{2 x^{1 + \frac{1}{2}}}$

Step 11.8.4

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

$\frac{2 x - 5}{2 x^{\frac{2}{2} + \frac{1}{2}}}$

Step 11.8.5

Combine the numerators over the common denominator.

$\frac{2 x - 5}{2 x^{\frac{2 + 1}{2}}}$

Step 11.8.6

Add $2$ and $1$ .

$\frac{2 x - 5}{2 x^{\frac{3}{2}}}$