Calculus Examples

y = \sqrt{4 x}

$y = \sqrt{4 x}$

Step 1

Rewrite

\frac{d}{d x} [\sqrt{4 x}]

$\frac{d}{d x} [\sqrt{4 x}]$ as

\frac{d}{d x} [2 \sqrt{x}]

$\frac{d}{d x} [2 \sqrt{x}]$ .

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Step 1.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

\frac{d}{d x} [\sqrt{2^{2} x}]

$\frac{d}{d x} [\sqrt{2^{2} x}]$

Step 1.2

Pull terms out from under the radical.

\frac{d}{d x} [2 \sqrt{x}]

$\frac{d}{d x} [2 \sqrt{x}]$

\frac{d}{d x} [2 \sqrt{x}]

$\frac{d}{d x} [2 \sqrt{x}]$

Step 2

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

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

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

\frac{d}{d x} [2 x^{\frac{1}{2}}]

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

Step 3

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

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

$2 x^{\frac{1}{2}}$ with respect to

x

$x$ is

2 \frac{d}{d x} [x^{\frac{1}{2}}]

$2 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

2 \frac{d}{d x} [x^{\frac{1}{2}}]

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

Step 4

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{1}{2}

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

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

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

Step 5

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 6

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 7

Combine the numerators over the common denominator.

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

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 8.2

Subtract

2

$2$ from

1

$1$ .

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

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

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

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

Step 9

Move the negative in front of the fraction.

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

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

Step 10

Combine

\frac{1}{2}

$\frac{1}{2}$ and

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

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

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

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

Step 11

Combine

2

$2$ and

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

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

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

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

Step 12

Move

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

$x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 13

Cancel the common factor.

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

Step 14

Rewrite the expression.

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