Calculus Examples

$f (x) = (\sqrt{x}) (\sqrt[3]{x})$

Step 1

Rewrite the expression using the least common index of $6$ .

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Step 1.1

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

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

Step 1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$ .

$\frac{d}{d x} [x^{\frac{3}{6}} \sqrt[3]{x}]$

Step 1.3

Rewrite $x^{\frac{3}{6}}$ as $\sqrt[6]{x^{3}}$ .

$\frac{d}{d x} [\sqrt[6]{x^{3}} \sqrt[3]{x}]$

Step 1.4

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

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

Step 1.5

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{2}{6}}$ .

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

Step 1.6

Rewrite $x^{\frac{2}{6}}$ as $\sqrt[6]{x^{2}}$ .

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

Step 2

Combine using the product rule for radicals.

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

Step 3

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.1

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

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

Step 3.2

Add $3$ and $2$ .

$\frac{d}{d x} [\sqrt[6]{x^{5}}]$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x^{5}}$ as $x^{\frac{5}{6}}$ .

$\frac{d}{d x} [x^{\frac{5}{6}}]$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{6}$ .

$\frac{5}{6} x^{\frac{5}{6} - 1}$

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5}{6} x^{\frac{5}{6} - 1 \cdot \frac{6}{6}}$

Step 7

Combine $- 1$ and $\frac{6}{6}$ .

$\frac{5}{6} x^{\frac{5}{6} + \frac{- 1 \cdot 6}{6}}$

Step 8

Combine the numerators over the common denominator.

$\frac{5}{6} x^{\frac{5 - 1 \cdot 6}{6}}$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $6$ .

$\frac{5}{6} x^{\frac{5 - 6}{6}}$

Step 9.2

Subtract $6$ from $5$ .

$\frac{5}{6} x^{\frac{- 1}{6}}$

Step 10

Move the negative in front of the fraction.

$\frac{5}{6} x^{- \frac{1}{6}}$

Step 11

Simplify.

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Step 11.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{5}{6} \cdot \frac{1}{x^{\frac{1}{6}}}$

Step 11.2

Multiply $\frac{5}{6}$ by $\frac{1}{x^{\frac{1}{6}}}$ .

$\frac{5}{6 x^{\frac{1}{6}}}$