Algebra Examples

$y = x^{2} \sqrt{x}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

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

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

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Step 4.1.1

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

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

Step 4.1.2

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 4.1.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Multiply $2$ by $2$ .

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

Step 4.1.5.2

Add $4$ and $1$ .

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

Step 4.2

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 4.3

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 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $5$ .

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

Step 4.7

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

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

Step 5

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

$y' = \frac{5 x^{\frac{3}{2}}}{2}$

Step 6

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

$\frac{d y}{d x} = \frac{5 x^{\frac{3}{2}}}{2}$