Calculus Examples

$x^{2} \cdot \sqrt{x}$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $2$ by $2$ .

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

Step 2.5.2

Add $4$ and $1$ .

$\frac{d}{d x} [x^{\frac{5}{2}}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$\frac{5}{2} x^{\frac{5}{2} - 1}$

Step 4

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

$\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}$

Step 5

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

$\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}$

Step 6

Combine the numerators over the common denominator.

$\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $5$ .

$\frac{5}{2} x^{\frac{3}{2}}$

Step 8

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$\frac{5 x^{\frac{3}{2}}}{2}$