Calculus Examples

$\sqrt[6]{x^{7}}$

Step 1

Rewrite $\frac{d}{d x} [\sqrt[6]{x^{7}}]$ as $\frac{d}{d x} [x \sqrt[6]{x}]$ .

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Step 1.1

Factor out $x^{6}$ .

$\frac{d}{d x} [\sqrt[6]{x^{6} x}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [x \sqrt[6]{x}]$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

$\frac{d}{d x} [x \cdot x^{\frac{1}{6}}]$

Step 3

Multiply $x$ by $x^{\frac{1}{6}}$ by adding the exponents.

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Step 3.1

Multiply $x$ by $x^{\frac{1}{6}}$ .

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Step 3.1.1

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

$\frac{d}{d x} [x^{1} x^{\frac{1}{6}}]$

Step 3.1.2

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

$\frac{d}{d x} [x^{1 + \frac{1}{6}}]$

Step 3.2

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

$\frac{d}{d x} [x^{\frac{6}{6} + \frac{1}{6}}]$

Step 3.3

Combine the numerators over the common denominator.

$\frac{d}{d x} [x^{\frac{6 + 1}{6}}]$

Step 3.4

Add $6$ and $1$ .

$\frac{d}{d x} [x^{\frac{7}{6}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{6}$ .

$\frac{7}{6} x^{\frac{7}{6} - 1}$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{7}{6} x^{\frac{7}{6} - 1 \cdot \frac{6}{6}}$

Step 6

Combine $- 1$ and $\frac{6}{6}$ .

$\frac{7}{6} x^{\frac{7}{6} + \frac{- 1 \cdot 6}{6}}$

Step 7

Combine the numerators over the common denominator.

$\frac{7}{6} x^{\frac{7 - 1 \cdot 6}{6}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $6$ .

$\frac{7}{6} x^{\frac{7 - 6}{6}}$

Step 8.2

Subtract $6$ from $7$ .

$\frac{7}{6} x^{\frac{1}{6}}$

Step 9

Combine $\frac{7}{6}$ and $x^{\frac{1}{6}}$ .

$\frac{7 x^{\frac{1}{6}}}{6}$