Calculus Examples

$f (x) = \frac{63}{\sqrt[3]{x}}$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d x} [\frac{63}{x^{\frac{1}{3}}}]$

Step 1.2

Since $63$ is constant with respect to $x$ , the derivative of $\frac{63}{x^{\frac{1}{3}}}$ with respect to $x$ is $63 \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}]$ .

$63 \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}]$

Step 1.3

Apply basic rules of exponents.

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Step 1.3.1

Rewrite $\frac{1}{x^{\frac{1}{3}}}$ as ${(x^{\frac{1}{3}})}^{- 1}$ .

$63 \frac{d}{d x} [{(x^{\frac{1}{3}})}^{- 1}]$

Step 1.3.2

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

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Step 1.3.2.1

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

$63 \frac{d}{d x} [x^{\frac{1}{3} \cdot - 1}]$

Step 1.3.2.2

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

$63 \frac{d}{d x} [x^{\frac{- 1}{3}}]$

Step 1.3.2.3

Move the negative in front of the fraction.

$63 \frac{d}{d x} [x^{- \frac{1}{3}}]$

Step 1.4

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

$63 (- \frac{1}{3} x^{- \frac{1}{3} - 1})$

Step 1.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$63 (- \frac{1}{3} x^{- \frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 1.6

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

$63 (- \frac{1}{3} x^{- \frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.7

Combine the numerators over the common denominator.

$63 (- \frac{1}{3} x^{\frac{- 1 - 1 \cdot 3}{3}})$

Step 1.8

Simplify the numerator.

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Step 1.8.1

Multiply $- 1$ by $3$ .

$63 (- \frac{1}{3} x^{\frac{- 1 - 3}{3}})$

Step 1.8.2

Subtract $3$ from $- 1$ .

$63 (- \frac{1}{3} x^{\frac{- 4}{3}})$

Step 1.9

Move the negative in front of the fraction.

$63 (- \frac{1}{3} x^{- \frac{4}{3}})$

Step 1.10

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

$63 (- \frac{x^{- \frac{4}{3}}}{3})$

Step 1.11

Multiply $- 1$ by $63$ .

$- 63 \frac{x^{- \frac{4}{3}}}{3}$

Step 1.12

Combine $- 63$ and $\frac{x^{- \frac{4}{3}}}{3}$ .

$\frac{- 63 x^{- \frac{4}{3}}}{3}$

Step 1.13

Move $x^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 63}{3 x^{\frac{4}{3}}}$

Step 1.14

Factor $3$ out of $- 63$ .

$\frac{3 \cdot - 21}{3 x^{\frac{4}{3}}}$

Step 1.15

Cancel the common factors.

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Step 1.15.1

Factor $3$ out of $3 x^{\frac{4}{3}}$ .

$\frac{3 \cdot - 21}{3 (x^{\frac{4}{3}})}$

Step 1.15.2

Cancel the common factor.

$\frac{3 \cdot - 21}{3 x^{\frac{4}{3}}}$

Step 1.15.3

Rewrite the expression.

$\frac{- 21}{x^{\frac{4}{3}}}$

Step 1.16

Move the negative in front of the fraction.

$f' (x) = - \frac{21}{x^{\frac{4}{3}}}$

Step 2

Find the second derivative.

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Step 2.1

Since $- 21$ is constant with respect to $x$ , the derivative of $- \frac{21}{x^{\frac{4}{3}}}$ with respect to $x$ is $- 21 \frac{d}{d x} [\frac{1}{x^{\frac{4}{3}}}]$ .

$- 21 \frac{d}{d x} [\frac{1}{x^{\frac{4}{3}}}]$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Rewrite $\frac{1}{x^{\frac{4}{3}}}$ as ${(x^{\frac{4}{3}})}^{- 1}$ .

$- 21 \frac{d}{d x} [{(x^{\frac{4}{3}})}^{- 1}]$

Step 2.2.2

Multiply the exponents in ${(x^{\frac{4}{3}})}^{- 1}$ .

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Step 2.2.2.1

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

$- 21 \frac{d}{d x} [x^{\frac{4}{3} \cdot - 1}]$

Step 2.2.2.2

Multiply $\frac{4}{3} \cdot - 1$ .

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Step 2.2.2.2.1

Combine $\frac{4}{3}$ and $- 1$ .

$- 21 \frac{d}{d x} [x^{\frac{4 \cdot - 1}{3}}]$

Step 2.2.2.2.2

Multiply $4$ by $- 1$ .

$- 21 \frac{d}{d x} [x^{\frac{- 4}{3}}]$

Step 2.2.2.3

Move the negative in front of the fraction.

$- 21 \frac{d}{d x} [x^{- \frac{4}{3}}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{4}{3}$ .

$- 21 (- \frac{4}{3} x^{- \frac{4}{3} - 1})$

Step 2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 21 (- \frac{4}{3} x^{- \frac{4}{3} - 1 \cdot \frac{3}{3}})$

Step 2.5

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

$- 21 (- \frac{4}{3} x^{- \frac{4}{3} + \frac{- 1 \cdot 3}{3}})$

Step 2.6

Combine the numerators over the common denominator.

$- 21 (- \frac{4}{3} x^{\frac{- 4 - 1 \cdot 3}{3}})$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $3$ .

$- 21 (- \frac{4}{3} x^{\frac{- 4 - 3}{3}})$

Step 2.7.2

Subtract $3$ from $- 4$ .

$- 21 (- \frac{4}{3} x^{\frac{- 7}{3}})$

Step 2.8

Move the negative in front of the fraction.

$- 21 (- \frac{4}{3} x^{- \frac{7}{3}})$

Step 2.9

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

$- 21 (- \frac{x^{- \frac{7}{3}} \cdot 4}{3})$

Step 2.10

Multiply $- 1$ by $- 21$ .

$21 \frac{x^{- \frac{7}{3}} \cdot 4}{3}$

Step 2.11

Combine $21$ and $\frac{x^{- \frac{7}{3}} \cdot 4}{3}$ .

$\frac{21 (x^{- \frac{7}{3}} \cdot 4)}{3}$

Step 2.12

Multiply.

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Step 2.12.1

Multiply $4$ by $21$ .

$\frac{84 x^{- \frac{7}{3}}}{3}$

Step 2.12.2

Move $x^{- \frac{7}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{84}{3 x^{\frac{7}{3}}}$

Step 2.13

Factor $3$ out of $84$ .

$\frac{3 \cdot 28}{3 x^{\frac{7}{3}}}$

Step 2.14

Cancel the common factors.

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Step 2.14.1

Factor $3$ out of $3 x^{\frac{7}{3}}$ .

$\frac{3 \cdot 28}{3 (x^{\frac{7}{3}})}$

Step 2.14.2

Cancel the common factor.

$\frac{3 \cdot 28}{3 x^{\frac{7}{3}}}$

Step 2.14.3

Rewrite the expression.

$f'' (x) = \frac{28}{x^{\frac{7}{3}}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $\frac{28}{x^{\frac{7}{3}}}$ .

$\frac{28}{x^{\frac{7}{3}}}$