Calculus Examples

$\sqrt{x^{7}}$

Step 1

Rewrite $\frac{d}{d x} [\sqrt{x^{7}}]$ as $\frac{d}{d x} [x^{3} \sqrt{x}]$ .

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Step 1.1

Rewrite $x^{7}$ as ${(x^{3})}^{2} x$ .

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Step 1.1.1

Factor out $x^{6}$ .

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

Step 1.1.2

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$\frac{d}{d x} [\sqrt{{(x^{3})}^{2} x}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [x^{3} \sqrt{x}]$

Step 2

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

$\frac{d}{d x} [x^{3} x^{\frac{1}{2}}]$

Step 3

Multiply $x^{3}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 3.1

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

$\frac{d}{d x} [x^{3 + \frac{1}{2}}]$

Step 3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{d}{d x} [x^{3 \cdot \frac{2}{2} + \frac{1}{2}}]$

Step 3.3

Combine $3$ and $\frac{2}{2}$ .

$\frac{d}{d x} [x^{\frac{3 \cdot 2}{2} + \frac{1}{2}}]$

Step 3.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [x^{\frac{3 \cdot 2 + 1}{2}}]$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $3$ by $2$ .

$\frac{d}{d x} [x^{\frac{6 + 1}{2}}]$

Step 3.5.2

Add $6$ and $1$ .

$\frac{d}{d x} [x^{\frac{7}{2}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{2}$ .

$\frac{7}{2} x^{\frac{7}{2} - 1}$

Step 5

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

$\frac{7}{2} x^{\frac{7}{2} - 1 \cdot \frac{2}{2}}$

Step 6

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

$\frac{7}{2} x^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}}$

Step 7

Combine the numerators over the common denominator.

$\frac{7}{2} x^{\frac{7 - 1 \cdot 2}{2}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$\frac{7}{2} x^{\frac{7 - 2}{2}}$

Step 8.2

Subtract $2$ from $7$ .

$\frac{7}{2} x^{\frac{5}{2}}$

Step 9

Combine $\frac{7}{2}$ and $x^{\frac{5}{2}}$ .

$\frac{7 x^{\frac{5}{2}}}{2}$