Calculus Examples

$y = \frac{4 x^{\frac{3}{2}}}{x}$

Step 1

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

$\frac{d}{d y} [4 x^{\frac{3}{2}} x^{- 1}]$

Step 2

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

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Step 2.1

Move $x^{- 1}$ .

$\frac{d}{d y} [4 (x^{- 1} x^{\frac{3}{2}})]$

Step 2.2

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

$\frac{d}{d y} [4 x^{- 1 + \frac{3}{2}}]$

Step 2.3

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

$\frac{d}{d y} [4 x^{- 1 \cdot \frac{2}{2} + \frac{3}{2}}]$

Step 2.4

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

$\frac{d}{d y} [4 x^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}}]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{d}{d y} [4 x^{\frac{- 1 \cdot 2 + 3}{2}}]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{d}{d y} [4 x^{\frac{- 2 + 3}{2}}]$

Step 2.6.2

Add $- 2$ and $3$ .

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

Step 3

Since $4 x^{\frac{1}{2}}$ is constant with respect to $y$ , the derivative of $4 x^{\frac{1}{2}}$ with respect to $y$ is $0$ .

$0$