Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

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 $g (x) = a x$ .

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Step 2.2.1

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

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

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 $n = \frac{1}{2}$ .

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $- 1$ by $2$ .

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

Step 2.8.2

Subtract $2$ from $1$ .

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

Step 2.9

Move the negative in front of the fraction.

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

Step 2.10

Multiply $a$ by $1$ .

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Since $a$ is constant with respect to $x$ , the derivative of $\frac{a}{{(a x)}^{\frac{1}{2}}}$ with respect to $x$ is $a \frac{d}{d x} [\frac{1}{{(a x)}^{\frac{1}{2}}}]$ .

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as ${(a x)}^{\frac{1}{2}}$ .

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

Step 3.4.2

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

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

Step 3.4.3

Replace all occurrences of $u_{2}$ with ${(a x)}^{\frac{1}{2}}$ .

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

Step 3.5

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 $g (x) = a x$ .

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Step 3.5.1

To apply the Chain Rule, set $u_{3}$ as $a x$ .

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

Step 3.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 3.5.3

Replace all occurrences of $u_{3}$ with $a x$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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Step 3.8.1

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

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

Step 3.8.2

Cancel the common factor of $2$ .

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Step 3.8.2.1

Factor $2$ out of $- 2$ .

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

Step 3.8.2.2

Cancel the common factor.

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

Step 3.8.2.3

Rewrite the expression.

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Combine the numerators over the common denominator.

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

Step 3.12

Simplify the numerator.

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Step 3.12.1

Multiply $- 1$ by $2$ .

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

Step 3.12.2

Subtract $2$ from $1$ .

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

Step 3.13

Move the negative in front of the fraction.

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

Step 3.14

Multiply $a$ by $1$ .

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

Step 3.15

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

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

Step 3.16

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

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

Step 3.17

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

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

Step 3.18

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

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

Step 3.19

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

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

Step 3.20

Cancel the common factor.

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

Step 3.21

Rewrite the expression.

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

Step 3.22

Combine $a$ and $\frac{1}{2 {(a x)}^{\frac{1}{2}} (x)}$ .

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

Step 4

Simplify.

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Step 4.1

Apply the product rule to $a x$ .

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

Step 4.2

Apply the product rule to $a x$ .

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

Step 4.3

Combine terms.

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Step 4.3.1

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

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

Step 4.3.2

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

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Step 4.3.2.1

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

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Step 4.3.2.1.1

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

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

Step 4.3.2.1.2

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

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

Step 4.3.2.2

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

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

Step 4.3.2.3

Combine the numerators over the common denominator.

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

Step 4.3.2.4

Subtract $1$ from $2$ .

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

Combine the numerators over the common denominator.

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

Step 4.3.7

Add $2$ and $1$ .

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

Step 4.3.8

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

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

Step 4.3.9

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

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Step 4.3.9.1

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

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Step 4.3.9.1.1

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

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

Step 4.3.9.1.2

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

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

Step 4.3.9.2

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

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

Step 4.3.9.3

Combine the numerators over the common denominator.

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

Step 4.3.9.4

Subtract $1$ from $2$ .

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