Calculus Examples

$y = \frac{5 x^{2}}{\sqrt{x}}$

Step 1

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

$\frac{d}{d x} [\frac{5 x^{2}}{x^{\frac{1}{2}}}]$

Step 2

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

$\frac{d}{d x} [5 x^{2} x^{- \frac{1}{2}}]$

Step 3

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

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Step 3.1

Move $x^{- \frac{1}{2}}$ .

$\frac{d}{d x} [5 (x^{- \frac{1}{2}} x^{2})]$

Step 3.2

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

$\frac{d}{d x} [5 x^{- \frac{1}{2} + 2}]$

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $2$ .

$\frac{d}{d x} [5 x^{\frac{- 1 + 4}{2}}]$

Step 3.6.2

Add $- 1$ and $4$ .

$\frac{d}{d x} [5 x^{\frac{3}{2}}]$

Step 4

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{3}{2}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$5 \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$5 (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 6

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

$5 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 7

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

$5 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 8

Combine the numerators over the common denominator.

$5 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

$5 (\frac{3}{2} x^{\frac{3 - 2}{2}})$

Step 9.2

Subtract $2$ from $3$ .

$5 (\frac{3}{2} x^{\frac{1}{2}})$

Step 10

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$5 \frac{3 x^{\frac{1}{2}}}{2}$

Step 11

Combine $5$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$\frac{5 (3 x^{\frac{1}{2}})}{2}$

Step 12

Multiply $3$ by $5$ .

$\frac{15 x^{\frac{1}{2}}}{2}$