Calculus Examples

$\sqrt{\frac{4 x^{2}}{6 + 2 x}}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Cancel the common factor of $4$ and $6 + 2 x$ .

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Step 1.1.1

Factor $2$ out of $4 x^{2}$ .

$\frac{d}{d x} [\sqrt{\frac{2 (2 x^{2})}{6 + 2 x}}]$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $6$ .

$\frac{d}{d x} [\sqrt{\frac{2 (2 x^{2})}{2 \cdot 3 + 2 x}}]$

Step 1.1.2.2

Factor $2$ out of $2 x$ .

$\frac{d}{d x} [\sqrt{\frac{2 (2 x^{2})}{2 \cdot 3 + 2 (x)}}]$

Step 1.1.2.3

Factor $2$ out of $2 \cdot 3 + 2 (x)$ .

$\frac{d}{d x} [\sqrt{\frac{2 (2 x^{2})}{2 \cdot (3 + x)}}]$

Step 1.1.2.4

Cancel the common factor.

$\frac{d}{d x} [\sqrt{\frac{2 (2 x^{2})}{2 \cdot (3 + x)}}]$

Step 1.1.2.5

Rewrite the expression.

$\frac{d}{d x} [\sqrt{\frac{2 x^{2}}{3 + x}}]$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{2 x^{2}}{3 + x}}$ as ${(\frac{2 x^{2}}{3 + x})}^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = \frac{2 x^{2}}{3 + x}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{2 x^{2}}{3 + x}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [\frac{2 x^{2}}{3 + x}]$

Step 2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{2 x^{2}}{3 + x}]$

Step 2.3

Replace all occurrences of $u$ with $\frac{2 x^{2}}{3 + x}$ .

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Step 2.3.1

Factor $2$ out of $4 x^{2}$ .

$\frac{1}{2} {(\frac{2 (2 x^{2})}{6 + 2 x})}^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{2 x^{2}}{3 + x}]$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $6$ .

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

Step 2.3.2.2

Factor $2$ out of $2 x$ .

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

Step 2.3.2.3

Factor $2$ out of $2 \cdot 3 + 2 (x)$ .

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

Step 2.3.2.4

Cancel the common factor.

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

Step 2.3.2.5

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Differentiate using the Constant Multiple Rule.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x^{2}}{3 + x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x^{2}}{3 + x}]$ .

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

Step 7.3

Simplify terms.

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Step 7.3.1

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

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

Step 7.3.2

Cancel the common factor of $2$ .

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Step 7.3.2.1

Cancel the common factor.

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

Step 7.3.2.2

Rewrite the expression.

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

Step 7.3.3

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

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

Step 8

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 $g (x) = 3 + x$ .

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

Step 9

Differentiate.

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Step 9.1

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

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

Step 9.2

Move $2$ to the left of $3 + x$ .

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

Step 9.3

By the Sum Rule, the derivative of $3 + x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [x]$ .

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

Step 9.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

${(\frac{2 x^{2}}{3 + x})}^{- \frac{1}{2}} \frac{2 (3 + x) x - x^{2} (0 + \frac{d}{d x} [x])}{{(3 + x)}^{2}}$

Step 9.5

Add $0$ and $\frac{d}{d x} [x]$ .

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

Step 9.6

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

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

Step 9.7

Multiply $- 1$ by $1$ .

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

Step 10

Simplify.

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Step 10.1

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 10.2

Apply the product rule to $\frac{3 + x}{2 x^{2}}$ .

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

Step 10.3

Apply the product rule to $2 x^{2}$ .

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

Step 10.4

Apply the distributive property.

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

Step 10.5

Apply the distributive property.

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

Step 10.6

Combine terms.

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Step 10.6.1

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

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Step 10.6.1.1

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

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

Step 10.6.1.2

Cancel the common factor of $2$ .

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Step 10.6.1.2.1

Cancel the common factor.

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

Step 10.6.1.2.2

Rewrite the expression.

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

Step 10.6.2

Simplify.

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

Step 10.6.3

Multiply $2$ by $3$ .

$\frac{{(3 + x)}^{\frac{1}{2}}}{2^{\frac{1}{2}} x} \cdot \frac{6 x + 2 x \cdot x - x^{2}}{{(3 + x)}^{2}}$

Step 10.6.4

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

$\frac{{(3 + x)}^{\frac{1}{2}}}{2^{\frac{1}{2}} x} \cdot \frac{6 x + 2 (x^{1} x) - x^{2}}{{(3 + x)}^{2}}$

Step 10.6.5

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

$\frac{{(3 + x)}^{\frac{1}{2}}}{2^{\frac{1}{2}} x} \cdot \frac{6 x + 2 (x^{1} x^{1}) - x^{2}}{{(3 + x)}^{2}}$

Step 10.6.6

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

$\frac{{(3 + x)}^{\frac{1}{2}}}{2^{\frac{1}{2}} x} \cdot \frac{6 x + 2 x^{1 + 1} - x^{2}}{{(3 + x)}^{2}}$

Step 10.6.7

Add $1$ and $1$ .

$\frac{{(3 + x)}^{\frac{1}{2}}}{2^{\frac{1}{2}} x} \cdot \frac{6 x + 2 x^{2} - x^{2}}{{(3 + x)}^{2}}$

Step 10.6.8

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{{(3 + x)}^{\frac{1}{2}}}{2^{\frac{1}{2}} x} \cdot \frac{6 x + x^{2}}{{(3 + x)}^{2}}$

Step 10.6.9

Multiply $\frac{{(3 + x)}^{\frac{1}{2}}}{2^{\frac{1}{2}} x}$ by $\frac{6 x + x^{2}}{{(3 + x)}^{2}}$ .

$\frac{{(3 + x)}^{\frac{1}{2}} (6 x + x^{2})}{2^{\frac{1}{2}} x {(3 + x)}^{2}}$

Step 10.6.10

Move ${(3 + x)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{6 x + x^{2}}{2^{\frac{1}{2}} x {(3 + x)}^{2} {(3 + x)}^{- \frac{1}{2}}}$

Step 10.6.11

Multiply ${(3 + x)}^{2}$ by ${(3 + x)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 10.6.11.1

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

$\frac{6 x + x^{2}}{2^{\frac{1}{2}} x ({(3 + x)}^{- \frac{1}{2}} {(3 + x)}^{2})}$

Step 10.6.11.2

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

$\frac{6 x + x^{2}}{2^{\frac{1}{2}} x {(3 + x)}^{- \frac{1}{2} + 2}}$

Step 10.6.11.3

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

$\frac{6 x + x^{2}}{2^{\frac{1}{2}} x {(3 + x)}^{- \frac{1}{2} + 2 \cdot \frac{2}{2}}}$

Step 10.6.11.4

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

$\frac{6 x + x^{2}}{2^{\frac{1}{2}} x {(3 + x)}^{- \frac{1}{2} + \frac{2 \cdot 2}{2}}}$

Step 10.6.11.5

Combine the numerators over the common denominator.

$\frac{6 x + x^{2}}{2^{\frac{1}{2}} x {(3 + x)}^{\frac{- 1 + 2 \cdot 2}{2}}}$

Step 10.6.11.6

Simplify the numerator.

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Step 10.6.11.6.1

Multiply $2$ by $2$ .

$\frac{6 x + x^{2}}{2^{\frac{1}{2}} x {(3 + x)}^{\frac{- 1 + 4}{2}}}$

Step 10.6.11.6.2

Add $- 1$ and $4$ .

$\frac{6 x + x^{2}}{2^{\frac{1}{2}} x {(3 + x)}^{\frac{3}{2}}}$

Step 10.7

Reorder terms.

$\frac{x^{2} + 6 x}{2^{\frac{1}{2}} x {(3 + x)}^{\frac{3}{2}}}$

Step 10.8

Factor $x$ out of $x^{2} + 6 x$ .

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Step 10.8.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x + 6 x}{2^{\frac{1}{2}} x {(3 + x)}^{\frac{3}{2}}}$

Step 10.8.2

Factor $x$ out of $6 x$ .

$\frac{x \cdot x + x \cdot 6}{2^{\frac{1}{2}} x {(3 + x)}^{\frac{3}{2}}}$

Step 10.8.3

Factor $x$ out of $x \cdot x + x \cdot 6$ .

$\frac{x (x + 6)}{2^{\frac{1}{2}} x {(3 + x)}^{\frac{3}{2}}}$

Step 10.9

Cancel the common factor.

$\frac{x (x + 6)}{2^{\frac{1}{2}} x {(3 + x)}^{\frac{3}{2}}}$

Step 10.10

Rewrite the expression.

$\frac{x + 6}{2^{\frac{1}{2}} {(3 + x)}^{\frac{3}{2}}}$