Calculus Examples

$y = \sqrt{\frac{x}{x + 3}}$

Step 1

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

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

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 $g (x) = \frac{x}{x + 3}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

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 $f (x) = x$ and $g (x) = x + 3$ .

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

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 $n = 1$ .

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

Step 9.2

Multiply $x + 3$ by $1$ .

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

Simplify terms.

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Step 9.6.1

Add $1$ and $0$ .

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

Step 9.6.2

Multiply $- 1$ by $1$ .

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

Step 9.6.3

Subtract $x$ from $x$ .

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

Step 9.6.4

Add $0$ and $3$ .

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

Step 9.6.5

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

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

Step 9.6.6

Move $2$ to the left of ${(x + 3)}^{2}$ .

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

Combine terms.

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Step 10.3.1

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

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

Step 10.3.2

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

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

Step 10.3.3

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

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Step 10.3.3.1

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

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

Step 10.3.3.2

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

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

Step 10.3.3.3

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

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

Step 10.3.3.4

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

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

Step 10.3.3.5

Combine the numerators over the common denominator.

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

Step 10.3.3.6

Simplify the numerator.

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Step 10.3.3.6.1

Multiply $2$ by $2$ .

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

Step 10.3.3.6.2

Add $- 1$ and $4$ .

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

Step 10.4

Reorder terms.

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