Calculus Examples

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

Step 1

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

$\frac{d}{d x} [\sqrt{\frac{x}{a}}] + \frac{d}{d x} [\sqrt{\frac{a}{x}}]$

Step 2

Evaluate $\frac{d}{d x} [\sqrt{\frac{x}{a}}]$ .

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $\frac{x}{a}$ .

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

Step 2.2.2

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{x}{a}] + \frac{d}{d x} [\sqrt{\frac{a}{x}}]$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $\frac{x}{a}$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $- 1$ by $2$ .

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

Step 2.8.2

Subtract $2$ from $1$ .

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

Step 2.9

Move the negative in front of the fraction.

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

Step 2.10

Multiply $\frac{1}{a}$ by $1$ .

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

Step 2.11

Multiply $\frac{1}{a}$ by $\frac{1}{2}$ .

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

Step 2.12

Move $2$ to the left of $a$ .

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

Step 3

Evaluate $\frac{d}{d x} [\sqrt{\frac{a}{x}}]$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $\frac{a}{x}$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u_{2}$ with $\frac{a}{x}$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Combine the numerators over the common denominator.

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

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $- 1$ by $2$ .

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

Step 3.9.2

Subtract $2$ from $1$ .

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

Step 3.10

Move the negative in front of the fraction.

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

Step 3.11

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

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

Step 3.12

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

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

Step 3.13

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine terms.

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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Step 4.5.3.1

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

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

Step 4.5.3.2

Multiply $a^{- \frac{1}{2}}$ by $a$ .

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Step 4.5.3.2.1

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

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

Step 4.5.3.2.2

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

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

Step 4.5.3.3

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

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

Step 4.5.3.4

Combine the numerators over the common denominator.

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

Step 4.5.3.5

Add $- 1$ and $2$ .

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

Step 4.5.4

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

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

Step 4.5.5

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

$\frac{1}{2 a^{\frac{1}{2}} x^{\frac{1}{2}}} - \frac{x^{\frac{1}{2}} a}{2 \cdot a^{\frac{1}{2}} x^{2}}$

Step 4.5.6

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

$\frac{1}{2 a^{\frac{1}{2}} x^{\frac{1}{2}}} - \frac{a}{2 a^{\frac{1}{2}} x^{2} x^{- \frac{1}{2}}}$

Step 4.5.7

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

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Step 4.5.7.1

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

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

Step 4.5.7.2

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

$\frac{1}{2 a^{\frac{1}{2}} x^{\frac{1}{2}}} - \frac{a}{2 a^{\frac{1}{2}} x^{- \frac{1}{2} + 2}}$

Step 4.5.7.3

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

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

Step 4.5.7.4

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

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

Step 4.5.7.5

Combine the numerators over the common denominator.

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

Step 4.5.7.6

Simplify the numerator.

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Step 4.5.7.6.1

Multiply $2$ by $2$ .

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

Step 4.5.7.6.2

Add $- 1$ and $4$ .

$\frac{1}{2 a^{\frac{1}{2}} x^{\frac{1}{2}}} - \frac{a}{2 a^{\frac{1}{2}} x^{\frac{3}{2}}}$

Step 4.5.8

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

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

Step 4.5.9

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

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Step 4.5.9.1

Multiply $a$ by $a^{- \frac{1}{2}}$ .

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Step 4.5.9.1.1

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

$\frac{1}{2 a^{\frac{1}{2}} x^{\frac{1}{2}}} - \frac{a^{1} a^{- \frac{1}{2}}}{2 x^{\frac{3}{2}}}$

Step 4.5.9.1.2

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

$\frac{1}{2 a^{\frac{1}{2}} x^{\frac{1}{2}}} - \frac{a^{1 - \frac{1}{2}}}{2 x^{\frac{3}{2}}}$

Step 4.5.9.2

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

$\frac{1}{2 a^{\frac{1}{2}} x^{\frac{1}{2}}} - \frac{a^{\frac{2}{2} - \frac{1}{2}}}{2 x^{\frac{3}{2}}}$

Step 4.5.9.3

Combine the numerators over the common denominator.

$\frac{1}{2 a^{\frac{1}{2}} x^{\frac{1}{2}}} - \frac{a^{\frac{2 - 1}{2}}}{2 x^{\frac{3}{2}}}$

Step 4.5.9.4

Subtract $1$ from $2$ .

$\frac{1}{2 a^{\frac{1}{2}} x^{\frac{1}{2}}} - \frac{a^{\frac{1}{2}}}{2 x^{\frac{3}{2}}}$