Calculus Examples

$y = \frac{\sqrt{x}}{x + 1}$

Step 1

Set $y$ as a function of $x$ .

$f (x) = \frac{\sqrt{x}}{x + 1}$

Step 2

Find the derivative.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

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

Step 2.7.2

Subtract $2$ from $1$ .

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

Step 2.8

Combine fractions.

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Step 2.8.1

Move the negative in front of the fraction.

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

Step 2.8.2

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

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

Step 2.8.3

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $1$ and $0$ .

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

Step 2.12.2

Multiply $- 1$ by $1$ .

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

Step 2.13

Simplify.

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Step 2.13.1

Apply the distributive property.

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

Step 2.13.2

Simplify the numerator.

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Step 2.13.2.1

Simplify each term.

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Step 2.13.2.1.1

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

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

Step 2.13.2.1.2

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

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

Step 2.13.2.1.3

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

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Step 2.13.2.1.3.1

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

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Step 2.13.2.1.3.1.1

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

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

Step 2.13.2.1.3.1.2

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

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

Step 2.13.2.1.3.2

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

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

Step 2.13.2.1.3.3

Combine the numerators over the common denominator.

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

Step 2.13.2.1.3.4

Subtract $1$ from $2$ .

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

Step 2.13.2.1.4

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

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

Step 2.13.2.2

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

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

Step 2.13.2.3

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

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

Step 2.13.2.4

Combine the numerators over the common denominator.

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

Step 2.13.2.5

Simplify each term.

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Step 2.13.2.5.1

Simplify the numerator.

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Step 2.13.2.5.1.1

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

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Step 2.13.2.5.1.1.1

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

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

Step 2.13.2.5.1.1.2

Multiply by $1$ .

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

Step 2.13.2.5.1.1.3

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

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

Step 2.13.2.5.1.1.4

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

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

Step 2.13.2.5.1.2

Multiply $- 1$ by $2$ .

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

Step 2.13.2.5.1.3

Subtract $2$ from $1$ .

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

Step 2.13.2.5.2

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

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

Step 2.13.2.5.3

Move the negative in front of the fraction.

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

Step 2.13.3

Combine terms.

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Step 2.13.3.1

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

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

Step 2.13.3.2

Combine.

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

Step 2.13.3.3

Apply the distributive property.

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

Step 2.13.3.4

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

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Step 2.13.3.4.1

Cancel the common factor.

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

Step 2.13.3.4.2

Rewrite the expression.

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

Step 2.13.3.5

Multiply $- 1$ by $2$ .

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

Step 2.13.3.6

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

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

Step 2.13.3.7

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

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

Step 2.13.3.8

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

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Step 2.13.3.8.1

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

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

Step 2.13.3.8.2

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

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

Step 2.13.3.8.3

Combine the numerators over the common denominator.

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

Step 2.13.3.8.4

Add $1$ and $1$ .

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

Step 2.13.3.8.5

Divide $2$ by $2$ .

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

Step 2.13.3.9

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

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

Step 2.13.3.10

Move $- 2$ to the left of $x$ .

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

Step 2.13.3.11

Cancel the common factor of $- 2$ and $2$ .

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Step 2.13.3.11.1

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

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

Step 2.13.3.11.2

Cancel the common factors.

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Step 2.13.3.11.2.1

Factor $2$ out of $2$ .

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

Step 2.13.3.11.2.2

Cancel the common factor.

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

Step 2.13.3.11.2.3

Rewrite the expression.

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

Step 2.13.3.11.2.4

Divide $- x$ by $1$ .

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

Step 2.13.4

Factor $- 1$ out of $- x$ .

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

Step 2.13.5

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

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

Step 2.13.6

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

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

Step 2.13.7

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

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

Step 2.13.8

Move the negative in front of the fraction.

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

Step 3

Set the derivative equal to $0$ then solve the equation $- \frac{x - 1}{2 x^{\frac{1}{2}} {(x + 1)}^{2}} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$x - 1 = 0$

Step 3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Solve the original function $f (x) = \frac{\sqrt{x}}{x + 1}$ at $x = 1$ .

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Step 4.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{\sqrt{1}}{(1) + 1}$

Step 4.2

Simplify the result.

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Step 4.2.1

Any root of $1$ is $1$ .

$f (1) = \frac{1}{1 + 1}$

Step 4.2.2

Add $1$ and $1$ .

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

Step 4.2.3

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 5

The horizontal tangent line on function $f (x) = \frac{\sqrt{x}}{x + 1}$ is $y = \frac{1}{2}$ .

$y = \frac{1}{2}$

Step 6