Calculus Examples

$f (x) = \sqrt[7]{9 x^{3}}$

Step 1

Simplify with factoring out.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{9 x^{3}}$ as ${(9 x^{3})}^{\frac{1}{7}}$ .

$\frac{d}{d x} [{(9 x^{3})}^{\frac{1}{7}}]$

Step 1.2

Factor $9$ out of $9 x^{3}$ .

$\frac{d}{d x} [{(9 (x^{3}))}^{\frac{1}{7}}]$

Step 1.3

Apply basic rules of exponents.

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Step 1.3.1

Apply the product rule to $9 (x^{3})$ .

$\frac{d}{d x} [9^{\frac{1}{7}} {(x^{3})}^{\frac{1}{7}}]$

Step 1.3.2

Multiply the exponents in ${(x^{3})}^{\frac{1}{7}}$ .

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Step 1.3.2.1

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

$\frac{d}{d x} [9^{\frac{1}{7}} x^{3 (\frac{1}{7})}]$

Step 1.3.2.2

Combine $3$ and $\frac{1}{7}$ .

$\frac{d}{d x} [9^{\frac{1}{7}} x^{\frac{3}{7}}]$

Step 2

Since $9^{\frac{1}{7}}$ is constant with respect to $x$ , the derivative of $9^{\frac{1}{7}} x^{\frac{3}{7}}$ with respect to $x$ is $9^{\frac{1}{7}} \frac{d}{d x} [x^{\frac{3}{7}}]$ .

$9^{\frac{1}{7}} \frac{d}{d x} [x^{\frac{3}{7}}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{7}$ .

$9^{\frac{1}{7}} (\frac{3}{7} x^{\frac{3}{7} - 1})$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$9^{\frac{1}{7}} (\frac{3}{7} x^{\frac{3}{7} - 1 \cdot \frac{7}{7}})$

Step 5

Combine $- 1$ and $\frac{7}{7}$ .

$9^{\frac{1}{7}} (\frac{3}{7} x^{\frac{3}{7} + \frac{- 1 \cdot 7}{7}})$

Step 6

Combine the numerators over the common denominator.

$9^{\frac{1}{7}} (\frac{3}{7} x^{\frac{3 - 1 \cdot 7}{7}})$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $7$ .

$9^{\frac{1}{7}} (\frac{3}{7} x^{\frac{3 - 7}{7}})$

Step 7.2

Subtract $7$ from $3$ .

$9^{\frac{1}{7}} (\frac{3}{7} x^{\frac{- 4}{7}})$

Step 8

Move the negative in front of the fraction.

$9^{\frac{1}{7}} (\frac{3}{7} x^{- \frac{4}{7}})$

Step 9

Combine $\frac{3}{7}$ and $x^{- \frac{4}{7}}$ .

$9^{\frac{1}{7}} \frac{3 x^{- \frac{4}{7}}}{7}$

Step 10

Combine $9^{\frac{1}{7}}$ and $\frac{3 x^{- \frac{4}{7}}}{7}$ .

$\frac{9^{\frac{1}{7}} (3 x^{- \frac{4}{7}})}{7}$

Step 11

Simplify the expression.

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Step 11.1

Rewrite $9$ as $3^{2}$ .

$\frac{3 \cdot {(3^{2})}^{\frac{1}{7}} x^{- \frac{4}{7}}}{7}$

Step 11.2

Multiply the exponents in ${(3^{2})}^{\frac{1}{7}}$ .

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Step 11.2.1

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

$\frac{3 \cdot 3^{2 (\frac{1}{7})} x^{- \frac{4}{7}}}{7}$

Step 11.2.2

Combine $2$ and $\frac{1}{7}$ .

$\frac{3 \cdot 3^{\frac{2}{7}} x^{- \frac{4}{7}}}{7}$

Step 12

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

$\frac{3^{1 + \frac{2}{7}} x^{- \frac{4}{7}}}{7}$

Step 13

Write $1$ as a fraction with a common denominator.

$\frac{3^{\frac{7}{7} + \frac{2}{7}} x^{- \frac{4}{7}}}{7}$

Step 14

Combine the numerators over the common denominator.

$\frac{3^{\frac{7 + 2}{7}} x^{- \frac{4}{7}}}{7}$

Step 15

Add $7$ and $2$ .

$\frac{3^{\frac{9}{7}} x^{- \frac{4}{7}}}{7}$

Step 16

Move $x^{- \frac{4}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3^{\frac{9}{7}}}{7 x^{\frac{4}{7}}}$