Algebra Examples

$\sqrt{\frac{48}{x}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{48}{x}}$ as ${(\frac{48}{x})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(\frac{48}{x})}^{\frac{1}{2}}]$

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 $g (x) = \frac{48}{x}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{48}{x}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [\frac{48}{x}]$

Step 2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{48}{x}]$

Step 2.3

Replace all occurrences of $u$ with $\frac{48}{x}$ .

$\frac{1}{2} {(\frac{48}{x})}^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{48}{x}]$

Step 3

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

$\frac{1}{2} {(\frac{48}{x})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\frac{48}{x}]$

Step 4

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

$\frac{1}{2} {(\frac{48}{x})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\frac{48}{x}]$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{2} {(\frac{48}{x})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [\frac{48}{x}]$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(\frac{48}{x})}^{\frac{1 - 2}{2}} \frac{d}{d x} [\frac{48}{x}]$

Step 6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(\frac{48}{x})}^{\frac{- 1}{2}} \frac{d}{d x} [\frac{48}{x}]$

Step 7

Move the negative in front of the fraction.

$\frac{1}{2} {(\frac{48}{x})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{48}{x}]$

Step 8

Since $48$ is constant with respect to $x$ , the derivative of $\frac{48}{x}$ with respect to $x$ is $48 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{1}{2} {(\frac{48}{x})}^{- \frac{1}{2}} (48 \frac{d}{d x} [\frac{1}{x}])$

Step 9

Simplify terms.

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Step 9.1

Combine $48$ and $\frac{1}{2}$ .

$\frac{48}{2} {(\frac{48}{x})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{1}{x}]$

Step 9.2

Cancel the common factor of $48$ and $2$ .

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Step 9.2.1

Factor $2$ out of $48$ .

$\frac{2 \cdot 24}{2} {(\frac{48}{x})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{1}{x}]$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 24}{2 (1)} {(\frac{48}{x})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{1}{x}]$

Step 9.2.2.2

Cancel the common factor.

$\frac{2 \cdot 24}{2 \cdot 1} {(\frac{48}{x})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{1}{x}]$

Step 9.2.2.3

Rewrite the expression.

$\frac{24}{1} {(\frac{48}{x})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{1}{x}]$

Step 9.2.2.4

Divide $24$ by $1$ .

$24 {(\frac{48}{x})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{1}{x}]$

Step 9.3

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$24 {(\frac{48}{x})}^{- \frac{1}{2}} \frac{d}{d x} [x^{- 1}]$

Step 10

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

$24 {(\frac{48}{x})}^{- \frac{1}{2}} (- x^{- 2})$

Step 11

Multiply $- 1$ by $24$ .

$- 24 {(\frac{48}{x})}^{- \frac{1}{2}} x^{- 2}$

Step 12

Simplify.

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Step 12.1

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

$- 24 {(\frac{x}{48})}^{\frac{1}{2}} x^{- 2}$

Step 12.2

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

$- 24 {(\frac{x}{48})}^{\frac{1}{2}} \frac{1}{x^{2}}$

Step 12.3

Apply the product rule to $\frac{x}{48}$ .

$- 24 \frac{x^{\frac{1}{2}}}{48^{\frac{1}{2}}} \frac{1}{x^{2}}$

Step 12.4

Combine terms.

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Step 12.4.1

Combine $- 24$ and $\frac{x^{\frac{1}{2}}}{48^{\frac{1}{2}}}$ .

$\frac{- 24 x^{\frac{1}{2}}}{48^{\frac{1}{2}}} \cdot \frac{1}{x^{2}}$

Step 12.4.2

Move the negative in front of the fraction.

$- \frac{24 x^{\frac{1}{2}}}{48^{\frac{1}{2}}} \cdot \frac{1}{x^{2}}$

Step 12.4.3

Multiply $\frac{1}{x^{2}}$ by $\frac{24 x^{\frac{1}{2}}}{48^{\frac{1}{2}}}$ .

$- \frac{24 x^{\frac{1}{2}}}{x^{2} \cdot 48^{\frac{1}{2}}}$

Step 12.4.4

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

$- \frac{24}{x^{2} \cdot 48^{\frac{1}{2}} x^{- \frac{1}{2}}}$

Step 12.4.5

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

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Step 12.4.5.1

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

$- \frac{24}{x^{- \frac{1}{2}} x^{2} \cdot 48^{\frac{1}{2}}}$

Step 12.4.5.2

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

$- \frac{24}{x^{- \frac{1}{2} + 2} \cdot 48^{\frac{1}{2}}}$

Step 12.4.5.3

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

$- \frac{24}{x^{- \frac{1}{2} + 2 \cdot \frac{2}{2}} \cdot 48^{\frac{1}{2}}}$

Step 12.4.5.4

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

$- \frac{24}{x^{- \frac{1}{2} + \frac{2 \cdot 2}{2}} \cdot 48^{\frac{1}{2}}}$

Step 12.4.5.5

Combine the numerators over the common denominator.

$- \frac{24}{x^{\frac{- 1 + 2 \cdot 2}{2}} \cdot 48^{\frac{1}{2}}}$

Step 12.4.5.6

Simplify the numerator.

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Step 12.4.5.6.1

Multiply $2$ by $2$ .

$- \frac{24}{x^{\frac{- 1 + 4}{2}} \cdot 48^{\frac{1}{2}}}$

Step 12.4.5.6.2

Add $- 1$ and $4$ .

$- \frac{24}{x^{\frac{3}{2}} \cdot 48^{\frac{1}{2}}}$