Calculus Examples

$f (x) = \frac{4}{\sqrt[3]{3 t^{2} + t}}$

Step 1

Factor $t$ out of $3 t^{2} + t$ .

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Step 1.1

Factor $t$ out of $3 t^{2}$ .

$\frac{d}{d x} [\frac{4}{\sqrt[3]{t (3 t) + t}}]$

Step 1.2

Raise $t$ to the power of $1$ .

$\frac{d}{d x} [\frac{4}{\sqrt[3]{t (3 t) + t^{1}}}]$

Step 1.3

Factor $t$ out of $t^{1}$ .

$\frac{d}{d x} [\frac{4}{\sqrt[3]{t (3 t) + t \cdot 1}}]$

Step 1.4

Factor $t$ out of $t (3 t) + t \cdot 1$ .

$\frac{d}{d x} [\frac{4}{\sqrt[3]{t (3 t + 1)}}]$

Step 2

Multiply $\frac{4}{\sqrt[3]{t (3 t + 1)}}$ by $\frac{{\sqrt[3]{t (3 t + 1)}}^{2}}{{\sqrt[3]{t (3 t + 1)}}^{2}}$ .

$\frac{d}{d x} [\frac{4}{\sqrt[3]{t (3 t + 1)}} \cdot \frac{{\sqrt[3]{t (3 t + 1)}}^{2}}{{\sqrt[3]{t (3 t + 1)}}^{2}}]$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{4}{\sqrt[3]{t (3 t + 1)}}$ by $\frac{{\sqrt[3]{t (3 t + 1)}}^{2}}{{\sqrt[3]{t (3 t + 1)}}^{2}}$ .

$\frac{d}{d x} [\frac{4 {\sqrt[3]{t (3 t + 1)}}^{2}}{\sqrt[3]{t (3 t + 1)} {\sqrt[3]{t (3 t + 1)}}^{2}}]$

Step 3.2

Raise $\sqrt[3]{t (3 t + 1)}$ to the power of $1$ .

$\frac{d}{d x} [\frac{4 {\sqrt[3]{t (3 t + 1)}}^{2}}{{\sqrt[3]{t (3 t + 1)}}^{1} {\sqrt[3]{t (3 t + 1)}}^{2}}]$

Step 3.3

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

$\frac{d}{d x} [\frac{4 {\sqrt[3]{t (3 t + 1)}}^{2}}{{\sqrt[3]{t (3 t + 1)}}^{1 + 2}}]$

Step 3.4

Add $1$ and $2$ .

$\frac{d}{d x} [\frac{4 {\sqrt[3]{t (3 t + 1)}}^{2}}{{\sqrt[3]{t (3 t + 1)}}^{3}}]$

Step 3.5

Rewrite ${\sqrt[3]{t (3 t + 1)}}^{3}$ as $t (3 t + 1)$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{t (3 t + 1)}$ as ${(t (3 t + 1))}^{\frac{1}{3}}$ .

$\frac{d}{d x} [\frac{4 {\sqrt[3]{t (3 t + 1)}}^{2}}{{({(t (3 t + 1))}^{\frac{1}{3}})}^{3}}]$

Step 3.5.2

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

$\frac{d}{d x} [\frac{4 {\sqrt[3]{t (3 t + 1)}}^{2}}{{(t (3 t + 1))}^{\frac{1}{3} \cdot 3}}]$

Step 3.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{d}{d x} [\frac{4 {\sqrt[3]{t (3 t + 1)}}^{2}}{{(t (3 t + 1))}^{\frac{3}{3}}}]$

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{d}{d x} [\frac{4 {\sqrt[3]{t (3 t + 1)}}^{2}}{{(t (3 t + 1))}^{\frac{3}{3}}}]$

Step 3.5.4.2

Rewrite the expression.

$\frac{d}{d x} [\frac{4 {\sqrt[3]{t (3 t + 1)}}^{2}}{{(t (3 t + 1))}^{1}}]$

Step 3.5.5

Simplify.

$\frac{d}{d x} [\frac{4 {\sqrt[3]{t (3 t + 1)}}^{2}}{t (3 t + 1)}]$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[3]{t (3 t + 1)}}^{2}$ as $\sqrt[3]{{(t (3 t + 1))}^{2}}$ .

$\frac{d}{d x} [\frac{4 \sqrt[3]{{(t (3 t + 1))}^{2}}}{t (3 t + 1)}]$

Step 4.2

Apply the product rule to $t (3 t + 1)$ .

$\frac{d}{d x} [\frac{4 \sqrt[3]{t^{2} {(3 t + 1)}^{2}}}{t (3 t + 1)}]$

Step 5

Since $\frac{4 \sqrt[3]{t^{2} {(3 t + 1)}^{2}}}{t (3 t + 1)}$ is constant with respect to $x$ , the derivative of $\frac{4 \sqrt[3]{t^{2} {(3 t + 1)}^{2}}}{t (3 t + 1)}$ with respect to $x$ is $0$ .

$0$