Calculus Examples

$y = \sqrt{2 p x}$

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 $\sqrt{2 p x}$ as ${(2 p x)}^{\frac{1}{2}}$ .

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

Step 1.2

Simplify with factoring out.

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Step 1.2.1

Factor $2$ out of $2 p x$ .

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

Step 1.2.2

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

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

Step 1.3

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

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

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 $g (x) = p x$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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Step 12.2.1

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

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

Step 12.2.2

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

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

Step 12.3

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

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

Step 12.4

Combine the numerators over the common denominator.

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

Step 12.5

Add $- 1$ and $2$ .

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

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

$\frac{p}{2^{\frac{1}{2}} {(p x)}^{\frac{1}{2}}}$

Step 17

Simplify.

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Step 17.1

Apply the product rule to $p x$ .

$\frac{p}{2^{\frac{1}{2}} (p^{\frac{1}{2}} x^{\frac{1}{2}})}$

Step 17.2

Combine terms.

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Step 17.2.1

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

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

Step 17.2.2

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

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Step 17.2.2.1

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

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Step 17.2.2.1.1

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

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

Step 17.2.2.1.2

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

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

Step 17.2.2.2

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

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

Step 17.2.2.3

Combine the numerators over the common denominator.

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

Step 17.2.2.4

Subtract $1$ from $2$ .

$\frac{p^{\frac{1}{2}}}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$