Calculus Examples

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

Step 1

Multiply $\frac{1}{2 \sqrt[3]{y^{2}}}$ by $\frac{{\sqrt[3]{y^{2}}}^{2}}{{\sqrt[3]{y^{2}}}^{2}}$ .

$\frac{d}{d \cdot d} [\frac{1}{2 \sqrt[3]{y^{2}}} \cdot \frac{{\sqrt[3]{y^{2}}}^{2}}{{\sqrt[3]{y^{2}}}^{2}}]$

Step 2

Combine and simplify the denominator.

Tap for more steps...

Step 2.1

Multiply $\frac{1}{2 \sqrt[3]{y^{2}}}$ by $\frac{{\sqrt[3]{y^{2}}}^{2}}{{\sqrt[3]{y^{2}}}^{2}}$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 \sqrt[3]{y^{2}} {\sqrt[3]{y^{2}}}^{2}}]$

Step 2.2

Move $\sqrt[3]{y^{2}}$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 (\sqrt[3]{y^{2}} {\sqrt[3]{y^{2}}}^{2})}]$

Step 2.3

Raise $\sqrt[3]{y^{2}}$ to the power of $1$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 ({\sqrt[3]{y^{2}}}^{1} {\sqrt[3]{y^{2}}}^{2})}]$

Step 2.4

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

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 {\sqrt[3]{y^{2}}}^{1 + 2}}]$

Step 2.5

Add $1$ and $2$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 {\sqrt[3]{y^{2}}}^{3}}]$

Step 2.6

Rewrite ${\sqrt[3]{y^{2}}}^{3}$ as $y^{2}$ .

Tap for more steps...

Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{2}}$ as $y^{\frac{2}{3}}$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 {(y^{\frac{2}{3}})}^{3}}]$

Step 2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 y^{\frac{2}{3} \cdot 3}}]$

Step 2.6.3

Combine $\frac{2}{3}$ and $3$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 y^{\frac{2 \cdot 3}{3}}}]$

Step 2.6.4

Multiply $2$ by $3$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 y^{\frac{6}{3}}}]$

Step 2.6.5

Cancel the common factor of $6$ and $3$ .

Tap for more steps...

Step 2.6.5.1

Factor $3$ out of $6$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 y^{\frac{3 \cdot 2}{3}}}]$

Step 2.6.5.2

Cancel the common factors.

Tap for more steps...

Step 2.6.5.2.1

Factor $3$ out of $3$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 y^{\frac{3 \cdot 2}{3 (1)}}}]$

Step 2.6.5.2.2

Cancel the common factor.

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 y^{\frac{3 \cdot 2}{3 \cdot 1}}}]$

Step 2.6.5.2.3

Rewrite the expression.

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 y^{\frac{2}{1}}}]$

Step 2.6.5.2.4

Divide $2$ by $1$ .

$\frac{d}{d \cdot d} [\frac{{\sqrt[3]{y^{2}}}^{2}}{2 y^{2}}]$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

Rewrite ${\sqrt[3]{y^{2}}}^{2}$ as $\sqrt[3]{{(y^{2})}^{2}}$ .

$\frac{d}{d \cdot d} [\frac{\sqrt[3]{{(y^{2})}^{2}}}{2 y^{2}}]$

Step 3.2

Multiply the exponents in ${(y^{2})}^{2}$ .

Tap for more steps...

Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d \cdot d} [\frac{\sqrt[3]{y^{2 \cdot 2}}}{2 y^{2}}]$

Step 3.2.2

Multiply $2$ by $2$ .

$\frac{d}{d \cdot d} [\frac{\sqrt[3]{y^{4}}}{2 y^{2}}]$

Step 3.3

Factor out $y^{3}$ .

$\frac{d}{d \cdot d} [\frac{\sqrt[3]{y^{3} y}}{2 y^{2}}]$

Step 3.4

Pull terms out from under the radical.

$\frac{d}{d \cdot d} [\frac{y \sqrt[3]{y}}{2 y^{2}}]$

Step 4

Cancel the common factor of $y$ and $y^{2}$ .

Tap for more steps...

Step 4.1

Factor $y$ out of $y \sqrt[3]{y}$ .

$\frac{d}{d \cdot d} [\frac{y (\sqrt[3]{y})}{2 y^{2}}]$

Step 4.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1

Factor $y$ out of $2 y^{2}$ .

$\frac{d}{d \cdot d} [\frac{y (\sqrt[3]{y})}{y (2 y)}]$

Step 4.2.2

Cancel the common factor.

$\frac{d}{d \cdot d} [\frac{y \sqrt[3]{y}}{y (2 y)}]$

Step 4.2.3

Rewrite the expression.

$\frac{d}{d \cdot d} [\frac{\sqrt[3]{y}}{2 y}]$

Step 5

Since $\frac{\sqrt[3]{y}}{2 y}$ is constant with respect to $d$ , the derivative of $\frac{\sqrt[3]{y}}{2 y}$ with respect to $d$ is $0$ .

$0$