Calculus Examples

$y = \sqrt{2 x}$

Step 1

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

$y = {(2 x)}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(2 x)}^{\frac{1}{2}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Simplify with factoring out.

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Step 4.1.1

Factor $2$ out of $2 x$ .

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

Step 4.1.2

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $2$ .

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

Step 4.7.2

Subtract $2$ from $1$ .

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

Step 4.8

Move the negative in front of the fraction.

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

Simplify the expression.

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Step 4.11.1

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

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

Step 4.11.2

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

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

Step 4.12

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

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Step 4.12.1

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

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Step 4.12.1.1

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

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

Step 4.12.1.2

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

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

Step 4.12.2

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

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

Step 4.12.3

Combine the numerators over the common denominator.

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

Step 4.12.4

Subtract $1$ from $2$ .

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

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$