Calculus Examples

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

Step 1

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

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

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{\frac{1}{2}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 3.5

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Simplify terms.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Cancel the common factor.

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

Step 8.6

Rewrite the expression.

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

Step 9

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

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

Step 10

Combine fractions.

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Step 10.1

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

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

Step 10.2

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

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Combine the numerators over the common denominator.

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

Step 12.5

Simplify the numerator.

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Step 12.5.1

Multiply $2$ by $2$ .

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

Step 12.5.2

Subtract $1$ from $4$ .

$\frac{2 (2 x^{\frac{1}{2}} + 1) x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

$\frac{(2 (2 x^{\frac{1}{2}}) + 2 \cdot 1) x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.2

Apply the distributive property.

$\frac{2 (2 x^{\frac{1}{2}}) x + 2 \cdot 1 x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.3

Simplify the numerator.

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Step 13.3.1

Simplify each term.

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Step 13.3.1.1

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

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Step 13.3.1.1.1

Move $x$ .

$\frac{2 (2 (x \cdot x^{\frac{1}{2}})) + 2 \cdot 1 x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.3.1.1.2

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

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Step 13.3.1.1.2.1

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

$\frac{2 (2 (x^{1} x^{\frac{1}{2}})) + 2 \cdot 1 x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.3.1.1.2.2

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

$\frac{2 (2 x^{1 + \frac{1}{2}}) + 2 \cdot 1 x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.3.1.1.3

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

$\frac{2 (2 x^{\frac{2}{2} + \frac{1}{2}}) + 2 \cdot 1 x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.3.1.1.4

Combine the numerators over the common denominator.

$\frac{2 (2 x^{\frac{2 + 1}{2}}) + 2 \cdot 1 x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.3.1.1.5

Add $2$ and $1$ .

$\frac{2 (2 x^{\frac{3}{2}}) + 2 \cdot 1 x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.3.1.2

Multiply $2$ by $2$ .

$\frac{4 x^{\frac{3}{2}} + 2 \cdot 1 x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.3.1.3

Multiply $2$ by $1$ .

$\frac{4 x^{\frac{3}{2}} + 2 x - x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.3.2

Subtract $x^{\frac{3}{2}}$ from $4 x^{\frac{3}{2}}$ .

$\frac{2 x + 3 x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.4

Factor $x$ out of $2 x + 3 x^{\frac{3}{2}}$ .

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Step 13.4.1

Factor $x$ out of $2 x$ .

$\frac{x \cdot 2 + 3 x^{\frac{3}{2}}}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.4.2

Factor $x$ out of $3 x^{\frac{3}{2}}$ .

$\frac{x \cdot 2 + x (3 x^{\frac{1}{2}})}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$

Step 13.4.3

Factor $x$ out of $x \cdot 2 + x (3 x^{\frac{1}{2}})$ .

$\frac{x (2 + 3 x^{\frac{1}{2}})}{{(2 x^{\frac{1}{2}} + 1)}^{2}}$