Calculus Examples

$\frac{1}{\sqrt{2 a x}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

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

Step 1.2

Simplify with factoring out.

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Step 1.2.1

Factor $2$ out of $2 a x$ .

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

Step 1.2.2

Apply the product rule to $2 (a x)$ .

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

Step 1.3

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

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

Step 1.4

Apply basic rules of exponents.

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Step 1.4.1

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

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

Step 1.4.2

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

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Step 1.4.2.1

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

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

Step 1.4.2.2

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

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

Step 1.4.2.3

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $a x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $a x$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $- 1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Combine ${(a x)}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Move $2$ .

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

Step 11.2

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

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Step 11.2.1

Raise $2$ to the power of $1$ .

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

Step 11.2.2

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

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

Step 11.3

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

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

Step 11.4

Combine the numerators over the common denominator.

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

Step 11.5

Add $2$ and $1$ .

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

Step 12

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

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

Step 13

Combine $x$ and $\frac{1}{2^{\frac{3}{2}} {(a x)}^{\frac{3}{2}}}$ .

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

Step 14

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

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

Step 15

Multiply $- 1$ by $1$ .

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

Step 16

Simplify.

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Step 16.1

Apply the product rule to $a x$ .

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

Step 16.2

Combine terms.

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Step 16.2.1

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

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

Step 16.2.2

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

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Step 16.2.2.1

Move $x^{- 1}$ .

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

Step 16.2.2.2

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

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

Step 16.2.2.3

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

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

Step 16.2.2.4

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

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

Step 16.2.2.5

Combine the numerators over the common denominator.

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

Step 16.2.2.6

Simplify the numerator.

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Step 16.2.2.6.1

Multiply $- 1$ by $2$ .

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

Step 16.2.2.6.2

Add $- 2$ and $3$ .

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