Calculus Examples

$y = \sqrt[4]{8 x^{2}}$

Step 1

Simplify with factoring out.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8 x^{2}}$ as ${(8 x^{2})}^{\frac{1}{4}}$ .

$\frac{d}{d x} [{(8 x^{2})}^{\frac{1}{4}}]$

Step 1.2

Factor $8$ out of $8 x^{2}$ .

$\frac{d}{d x} [{(8 (x^{2}))}^{\frac{1}{4}}]$

Step 1.3

Apply basic rules of exponents.

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Step 1.3.1

Apply the product rule to $8 (x^{2})$ .

$\frac{d}{d x} [8^{\frac{1}{4}} {(x^{2})}^{\frac{1}{4}}]$

Step 1.3.2

Multiply the exponents in ${(x^{2})}^{\frac{1}{4}}$ .

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Step 1.3.2.1

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

$\frac{d}{d x} [8^{\frac{1}{4}} x^{2 (\frac{1}{4})}]$

Step 1.3.2.2

Cancel the common factor of $2$ .

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Step 1.3.2.2.1

Factor $2$ out of $4$ .

$\frac{d}{d x} [8^{\frac{1}{4}} x^{2 \frac{1}{2 (2)}}]$

Step 1.3.2.2.2

Cancel the common factor.

$\frac{d}{d x} [8^{\frac{1}{4}} x^{2 \frac{1}{2 \cdot 2}}]$

Step 1.3.2.2.3

Rewrite the expression.

$\frac{d}{d x} [8^{\frac{1}{4}} x^{\frac{1}{2}}]$

Step 2

Since $8^{\frac{1}{4}}$ is constant with respect to $x$ , the derivative of $8^{\frac{1}{4}} x^{\frac{1}{2}}$ with respect to $x$ is $8^{\frac{1}{4}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$8^{\frac{1}{4}} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3

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

$8^{\frac{1}{4}} (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 4

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

$8^{\frac{1}{4}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 5

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

$8^{\frac{1}{4}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 6

Combine the numerators over the common denominator.

$8^{\frac{1}{4}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$8^{\frac{1}{4}} (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 7.2

Subtract $2$ from $1$ .

$8^{\frac{1}{4}} (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 8

Move the negative in front of the fraction.

$8^{\frac{1}{4}} (\frac{1}{2} x^{- \frac{1}{2}})$

Step 9

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

$8^{\frac{1}{4}} \frac{x^{- \frac{1}{2}}}{2}$

Step 10

Combine $8^{\frac{1}{4}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{8^{\frac{1}{4}} x^{- \frac{1}{2}}}{2}$

Step 11

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

$\frac{8^{\frac{1}{4}}}{2 x^{\frac{1}{2}}}$