Calculus Examples

$y = \frac{6 x^{\frac{3}{2}}}{x}$

Step 1

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

$\frac{d}{d x} [6 x^{\frac{3}{2}} x^{- 1}]$

Step 2

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

Tap for more steps...

Step 2.1

Move $x^{- 1}$ .

$\frac{d}{d x} [6 (x^{- 1} x^{\frac{3}{2}})]$

Step 2.2

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

$\frac{d}{d x} [6 x^{- 1 + \frac{3}{2}}]$

Step 2.3

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

$\frac{d}{d x} [6 x^{- 1 \cdot \frac{2}{2} + \frac{3}{2}}]$

Step 2.4

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

$\frac{d}{d x} [6 x^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}}]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{d}{d x} [6 x^{\frac{- 1 \cdot 2 + 3}{2}}]$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{d}{d x} [6 x^{\frac{- 2 + 3}{2}}]$

Step 2.6.2

Add $- 2$ and $3$ .

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

Step 3

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{\frac{1}{2}}$ with respect to $x$ is $6 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 4

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

$6 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 5

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

$6 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 6

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

$6 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 7

Combine the numerators over the common denominator.

$6 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Multiply $- 1$ by $2$ .

$6 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 8.2

Subtract $2$ from $1$ .

$6 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 9

Move the negative in front of the fraction.

$6 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 10

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

$6 \frac{x^{- \frac{1}{2}}}{2}$

Step 11

Combine $6$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{6 x^{- \frac{1}{2}}}{2}$

Step 12

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

$\frac{6}{2 x^{\frac{1}{2}}}$

Step 13

Factor $2$ out of $6$ .

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

Step 14

Cancel the common factors.

Tap for more steps...

Step 14.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 14.2

Cancel the common factor.

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

Step 14.3

Rewrite the expression.

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