Calculus Examples

$y = \sqrt{\frac{x}{x + 8}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x}{x + 8}}$ as ${(\frac{x}{x + 8})}^{\frac{1}{2}}$ .

$y = {(\frac{x}{x + 8})}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(\frac{x}{x + 8})}^{\frac{1}{2}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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 $g (x) = \frac{x}{x + 8}$ .

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Step 4.1.1

To apply the Chain Rule, set $u$ as $\frac{x}{x + 8}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [\frac{x}{x + 8}]$

Step 4.1.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{x}{x + 8}]$

Step 4.1.3

Replace all occurrences of $u$ with $\frac{x}{x + 8}$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{x}{x + 8}]$

Step 4.2

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

$\frac{1}{2} {(\frac{x}{x + 8})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\frac{x}{x + 8}]$

Step 4.3

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

$\frac{1}{2} {(\frac{x}{x + 8})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\frac{x}{x + 8}]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{1}{2} {(\frac{x}{x + 8})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [\frac{x}{x + 8}]$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{\frac{1 - 2}{2}} \frac{d}{d x} [\frac{x}{x + 8}]$

Step 4.5.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{\frac{- 1}{2}} \frac{d}{d x} [\frac{x}{x + 8}]$

Step 4.6

Move the negative in front of the fraction.

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{x}{x + 8}]$

Step 4.7

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 $f (x) = x$ and $g (x) = x + 8$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{(x + 8) \frac{d}{d x} [x] - x \frac{d}{d x} [x + 8]}{{(x + 8)}^{2}}$

Step 4.8

Differentiate.

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Step 4.8.1

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

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{(x + 8) \cdot 1 - x \frac{d}{d x} [x + 8]}{{(x + 8)}^{2}}$

Step 4.8.2

Multiply $x + 8$ by $1$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{x + 8 - x \frac{d}{d x} [x + 8]}{{(x + 8)}^{2}}$

Step 4.8.3

By the Sum Rule, the derivative of $x + 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [8]$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{x + 8 - x (\frac{d}{d x} [x] + \frac{d}{d x} [8])}{{(x + 8)}^{2}}$

Step 4.8.4

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

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{x + 8 - x (1 + \frac{d}{d x} [8])}{{(x + 8)}^{2}}$

Step 4.8.5

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{x + 8 - x (1 + 0)}{{(x + 8)}^{2}}$

Step 4.8.6

Simplify terms.

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Step 4.8.6.1

Add $1$ and $0$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{x + 8 - x \cdot 1}{{(x + 8)}^{2}}$

Step 4.8.6.2

Multiply $- 1$ by $1$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{x + 8 - x}{{(x + 8)}^{2}}$

Step 4.8.6.3

Subtract $x$ from $x$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{0 + 8}{{(x + 8)}^{2}}$

Step 4.8.6.4

Add $0$ and $8$ .

$\frac{1}{2} {(\frac{x}{x + 8})}^{- \frac{1}{2}} \frac{8}{{(x + 8)}^{2}}$

Step 4.8.6.5

Multiply $\frac{8}{{(x + 8)}^{2}}$ by $\frac{1}{2}$ .

$\frac{8}{{(x + 8)}^{2} \cdot 2} {(\frac{x}{x + 8})}^{- \frac{1}{2}}$

Step 4.8.6.6

Move $2$ to the left of ${(x + 8)}^{2}$ .

$\frac{8}{2 \cdot {(x + 8)}^{2}} {(\frac{x}{x + 8})}^{- \frac{1}{2}}$

Step 4.8.6.7

Cancel the common factor of $8$ and $2$ .

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Step 4.8.6.7.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 4}{2 {(x + 8)}^{2}} {(\frac{x}{x + 8})}^{- \frac{1}{2}}$

Step 4.8.6.7.2

Cancel the common factors.

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Step 4.8.6.7.2.1

Factor $2$ out of $2 {(x + 8)}^{2}$ .

$\frac{2 \cdot 4}{2 ({(x + 8)}^{2})} {(\frac{x}{x + 8})}^{- \frac{1}{2}}$

Step 4.8.6.7.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 {(x + 8)}^{2}} {(\frac{x}{x + 8})}^{- \frac{1}{2}}$

Step 4.8.6.7.2.3

Rewrite the expression.

$\frac{4}{{(x + 8)}^{2}} {(\frac{x}{x + 8})}^{- \frac{1}{2}}$

Step 4.9

Simplify.

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Step 4.9.1

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

$\frac{4}{{(x + 8)}^{2}} {(\frac{x + 8}{x})}^{\frac{1}{2}}$

Step 4.9.2

Apply the product rule to $\frac{x + 8}{x}$ .

$\frac{4}{{(x + 8)}^{2}} \cdot \frac{{(x + 8)}^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 4.9.3

Combine terms.

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Step 4.9.3.1

Multiply $\frac{4}{{(x + 8)}^{2}}$ by $\frac{{(x + 8)}^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

$\frac{4 {(x + 8)}^{\frac{1}{2}}}{{(x + 8)}^{2} x^{\frac{1}{2}}}$

Step 4.9.3.2

Move ${(x + 8)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{4}{{(x + 8)}^{2} x^{\frac{1}{2}} {(x + 8)}^{- \frac{1}{2}}}$

Step 4.9.3.3

Multiply ${(x + 8)}^{2}$ by ${(x + 8)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.9.3.3.1

Move ${(x + 8)}^{- \frac{1}{2}}$ .

$\frac{4}{{(x + 8)}^{- \frac{1}{2}} {(x + 8)}^{2} x^{\frac{1}{2}}}$

Step 4.9.3.3.2

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

$\frac{4}{{(x + 8)}^{- \frac{1}{2} + 2} x^{\frac{1}{2}}}$

Step 4.9.3.3.3

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

$\frac{4}{{(x + 8)}^{- \frac{1}{2} + 2 \cdot \frac{2}{2}} x^{\frac{1}{2}}}$

Step 4.9.3.3.4

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

$\frac{4}{{(x + 8)}^{- \frac{1}{2} + \frac{2 \cdot 2}{2}} x^{\frac{1}{2}}}$

Step 4.9.3.3.5

Combine the numerators over the common denominator.

$\frac{4}{{(x + 8)}^{\frac{- 1 + 2 \cdot 2}{2}} x^{\frac{1}{2}}}$

Step 4.9.3.3.6

Simplify the numerator.

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Step 4.9.3.3.6.1

Multiply $2$ by $2$ .

$\frac{4}{{(x + 8)}^{\frac{- 1 + 4}{2}} x^{\frac{1}{2}}}$

Step 4.9.3.3.6.2

Add $- 1$ and $4$ .

$\frac{4}{{(x + 8)}^{\frac{3}{2}} x^{\frac{1}{2}}}$

Step 4.9.4

Reorder terms.

$\frac{4}{x^{\frac{1}{2}} {(x + 8)}^{\frac{3}{2}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{4}{x^{\frac{1}{2}} {(x + 8)}^{\frac{3}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{4}{x^{\frac{1}{2}} {(x + 8)}^{\frac{3}{2}}}$