Calculus Examples

$y = \sqrt[3]{x}$

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 $x^{\frac{1}{3}}$ .

$f' (x) = \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.2

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

$f' (x) = \frac{1}{3} x^{\frac{1}{3} - 1}$

Step 1.3

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

$f' (x) = \frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}$

Step 1.4

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

$f' (x) = \frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}$

Step 1.5

Combine the numerators over the common denominator.

$f' (x) = \frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $3$ .

$f' (x) = \frac{1}{3} x^{\frac{1 - 3}{3}}$

Step 1.6.2

Subtract $3$ from $1$ .

$f' (x) = \frac{1}{3} x^{\frac{- 2}{3}}$

Step 1.7

Move the negative in front of the fraction.

$f' (x) = \frac{1}{3} x^{- \frac{2}{3}}$

Step 1.8

Simplify.

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Step 1.8.1

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

$f' (x) = \frac{1}{3} \cdot \frac{1}{x^{\frac{2}{3}}}$

Step 1.8.2

Multiply $\frac{1}{3}$ by $\frac{1}{x^{\frac{2}{3}}}$ .

$f' (x) = \frac{1}{3 x^{\frac{2}{3}}}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3 x^{\frac{2}{3}}}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}]$ .

$f'' (x) = \frac{1}{3} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{2}{3}}})$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Rewrite $\frac{1}{x^{\frac{2}{3}}}$ as ${(x^{\frac{2}{3}})}^{- 1}$ .

$f'' (x) = \frac{1}{3} \cdot \frac{d}{d x} ({(x^{\frac{2}{3}})}^{- 1})$

Step 2.2.2

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

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Step 2.2.2.1

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

$f'' (x) = \frac{1}{3} \cdot \frac{d}{d x} (x^{\frac{2}{3} \cdot - 1})$

Step 2.2.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 2.2.2.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$f'' (x) = \frac{1}{3} \cdot \frac{d}{d x} (x^{\frac{2 \cdot - 1}{3}})$

Step 2.2.2.2.2

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{1}{3} \cdot \frac{d}{d x} (x^{\frac{- 2}{3}})$

Step 2.2.2.3

Move the negative in front of the fraction.

$f'' (x) = \frac{1}{3} \cdot \frac{d}{d x} (x^{- \frac{2}{3}})$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{2}{3}$ .

$f'' (x) = \frac{1}{3} \cdot (- \frac{2}{3} x^{- \frac{2}{3} - 1})$

Step 2.4

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

$f'' (x) = \frac{1}{3} \cdot (- \frac{2}{3} x^{- \frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 2.5

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

$f'' (x) = \frac{1}{3} \cdot (- \frac{2}{3} x^{- \frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 2.6

Combine the numerators over the common denominator.

$f'' (x) = \frac{1}{3} \cdot (- \frac{2}{3} x^{\frac{- 2 - 1 \cdot 3}{3}})$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $3$ .

$f'' (x) = \frac{1}{3} \cdot (- \frac{2}{3} x^{\frac{- 2 - 3}{3}})$

Step 2.7.2

Subtract $3$ from $- 2$ .

$f'' (x) = \frac{1}{3} \cdot (- \frac{2}{3} x^{\frac{- 5}{3}})$

Step 2.8

Move the negative in front of the fraction.

$f'' (x) = \frac{1}{3} \cdot (- \frac{2}{3} x^{- \frac{5}{3}})$

Step 2.9

Combine $x^{- \frac{5}{3}}$ and $\frac{2}{3}$ .

$f'' (x) = \frac{1}{3} \cdot (- \frac{x^{- \frac{5}{3}} \cdot 2}{3})$

Step 2.10

Multiply $\frac{1}{3}$ by $\frac{x^{- \frac{5}{3}} \cdot 2}{3}$ .

$f'' (x) = - \frac{x^{- \frac{5}{3}} \cdot 2}{3 \cdot 3}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Multiply $3$ by $3$ .

$f'' (x) = - \frac{x^{- \frac{5}{3}} \cdot 2}{9}$

Step 2.11.2

Move $2$ to the left of $x^{- \frac{5}{3}}$ .

$f'' (x) = - \frac{2 \cdot x^{- \frac{5}{3}}}{9}$

Step 2.11.3

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{2}{9 x^{\frac{5}{3}}}$

Step 3

Find the third derivative.

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Step 3.1

Since $- \frac{2}{9}$ is constant with respect to $x$ , the derivative of $- \frac{2}{9 x^{\frac{5}{3}}}$ with respect to $x$ is $- \frac{2}{9} \frac{d}{d x} [\frac{1}{x^{\frac{5}{3}}}]$ .

$f''' (x) = - \frac{2}{9} \cdot \frac{d}{d x} \frac{1}{x^{\frac{5}{3}}}$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Rewrite $\frac{1}{x^{\frac{5}{3}}}$ as ${(x^{\frac{5}{3}})}^{- 1}$ .

$f''' (x) = - \frac{2}{9} \cdot \frac{d}{d x} {(x^{\frac{5}{3}})}^{- 1}$

Step 3.2.2

Multiply the exponents in ${(x^{\frac{5}{3}})}^{- 1}$ .

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Step 3.2.2.1

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

$f''' (x) = - \frac{2}{9} \cdot \frac{d}{d x} x^{\frac{5}{3} \cdot - 1}$

Step 3.2.2.2

Multiply $\frac{5}{3} \cdot - 1$ .

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Step 3.2.2.2.1

Combine $\frac{5}{3}$ and $- 1$ .

$f''' (x) = - \frac{2}{9} \cdot \frac{d}{d x} x^{\frac{5 \cdot - 1}{3}}$

Step 3.2.2.2.2

Multiply $5$ by $- 1$ .

$f''' (x) = - \frac{2}{9} \cdot \frac{d}{d x} x^{\frac{- 5}{3}}$

Step 3.2.2.3

Move the negative in front of the fraction.

$f''' (x) = - \frac{2}{9} \cdot \frac{d}{d x} x^{- \frac{5}{3}}$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{5}{3}$ .

$f''' (x) = - \frac{2}{9} \cdot (- \frac{5}{3} x^{- \frac{5}{3} - 1})$

Step 3.4

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

$f''' (x) = - \frac{2}{9} \cdot (- \frac{5}{3} x^{- \frac{5}{3} - 1 \cdot \frac{3}{3}})$

Step 3.5

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

$f''' (x) = - \frac{2}{9} \cdot (- \frac{5}{3} x^{- \frac{5}{3} + \frac{- 1 \cdot 3}{3}})$

Step 3.6

Combine the numerators over the common denominator.

$f''' (x) = - \frac{2}{9} \cdot (- \frac{5}{3} x^{\frac{- 5 - 1 \cdot 3}{3}})$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $3$ .

$f''' (x) = - \frac{2}{9} \cdot (- \frac{5}{3} x^{\frac{- 5 - 3}{3}})$

Step 3.7.2

Subtract $3$ from $- 5$ .

$f''' (x) = - \frac{2}{9} \cdot (- \frac{5}{3} x^{\frac{- 8}{3}})$

Step 3.8

Move the negative in front of the fraction.

$f''' (x) = - \frac{2}{9} \cdot (- \frac{5}{3} x^{- \frac{8}{3}})$

Step 3.9

Combine $x^{- \frac{8}{3}}$ and $\frac{5}{3}$ .

$f''' (x) = - \frac{2}{9} \cdot (- \frac{x^{- \frac{8}{3}} \cdot 5}{3})$

Step 3.10

Multiply.

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Step 3.10.1

Multiply $- 1$ by $- 1$ .

$f''' (x) = 1 (\frac{2}{9}) \cdot \frac{x^{- \frac{8}{3}} \cdot 5}{3}$

Step 3.10.2

Multiply $\frac{2}{9}$ by $1$ .

$f''' (x) = \frac{2}{9} \cdot \frac{x^{- \frac{8}{3}} \cdot 5}{3}$

Step 3.11

Multiply $\frac{2}{9}$ by $\frac{x^{- \frac{8}{3}} \cdot 5}{3}$ .

$f''' (x) = \frac{2 (x^{- \frac{8}{3}} \cdot 5)}{9 \cdot 3}$

Step 3.12

Multiply.

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Step 3.12.1

Multiply $5$ by $2$ .

$f''' (x) = \frac{10 x^{- \frac{8}{3}}}{9 \cdot 3}$

Step 3.12.2

Multiply $9$ by $3$ .

$f''' (x) = \frac{10 x^{- \frac{8}{3}}}{27}$

Step 3.12.3

Move $x^{- \frac{8}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f''' (x) = \frac{10}{27 x^{\frac{8}{3}}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $\frac{10}{27}$ is constant with respect to $x$ , the derivative of $\frac{10}{27 x^{\frac{8}{3}}}$ with respect to $x$ is $\frac{10}{27} \frac{d}{d x} [\frac{1}{x^{\frac{8}{3}}}]$ .

$f^{4} (x) = \frac{10}{27} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{8}{3}}})$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

Rewrite $\frac{1}{x^{\frac{8}{3}}}$ as ${(x^{\frac{8}{3}})}^{- 1}$ .

$f^{4} (x) = \frac{10}{27} \cdot \frac{d}{d x} ({(x^{\frac{8}{3}})}^{- 1})$

Step 4.2.2

Multiply the exponents in ${(x^{\frac{8}{3}})}^{- 1}$ .

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Step 4.2.2.1

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

$f^{4} (x) = \frac{10}{27} \cdot \frac{d}{d x} (x^{\frac{8}{3} \cdot - 1})$

Step 4.2.2.2

Multiply $\frac{8}{3} \cdot - 1$ .

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Step 4.2.2.2.1

Combine $\frac{8}{3}$ and $- 1$ .

$f^{4} (x) = \frac{10}{27} \cdot \frac{d}{d x} (x^{\frac{8 \cdot - 1}{3}})$

Step 4.2.2.2.2

Multiply $8$ by $- 1$ .

$f^{4} (x) = \frac{10}{27} \cdot \frac{d}{d x} (x^{\frac{- 8}{3}})$

Step 4.2.2.3

Move the negative in front of the fraction.

$f^{4} (x) = \frac{10}{27} \cdot \frac{d}{d x} (x^{- \frac{8}{3}})$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{8}{3}$ .

$f^{4} (x) = \frac{10}{27} \cdot (- \frac{8}{3} x^{- \frac{8}{3} - 1})$

Step 4.4

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

$f^{4} (x) = \frac{10}{27} \cdot (- \frac{8}{3} x^{- \frac{8}{3} - 1 \cdot \frac{3}{3}})$

Step 4.5

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

$f^{4} (x) = \frac{10}{27} \cdot (- \frac{8}{3} x^{- \frac{8}{3} + \frac{- 1 \cdot 3}{3}})$

Step 4.6

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{10}{27} \cdot (- \frac{8}{3} x^{\frac{- 8 - 1 \cdot 3}{3}})$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $3$ .

$f^{4} (x) = \frac{10}{27} \cdot (- \frac{8}{3} x^{\frac{- 8 - 3}{3}})$

Step 4.7.2

Subtract $3$ from $- 8$ .

$f^{4} (x) = \frac{10}{27} \cdot (- \frac{8}{3} x^{\frac{- 11}{3}})$

Step 4.8

Move the negative in front of the fraction.

$f^{4} (x) = \frac{10}{27} \cdot (- \frac{8}{3} x^{- \frac{11}{3}})$

Step 4.9

Combine $x^{- \frac{11}{3}}$ and $\frac{8}{3}$ .

$f^{4} (x) = \frac{10}{27} \cdot (- \frac{x^{- \frac{11}{3}} \cdot 8}{3})$

Step 4.10

Multiply $\frac{10}{27}$ by $\frac{x^{- \frac{11}{3}} \cdot 8}{3}$ .

$f^{4} (x) = - \frac{10 (x^{- \frac{11}{3}} \cdot 8)}{27 \cdot 3}$

Step 4.11

Multiply.

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Step 4.11.1

Multiply $8$ by $10$ .

$f^{4} (x) = - \frac{80 x^{- \frac{11}{3}}}{27 \cdot 3}$

Step 4.11.2

Multiply $27$ by $3$ .

$f^{4} (x) = - \frac{80 x^{- \frac{11}{3}}}{81}$

Step 4.11.3

Move $x^{- \frac{11}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{4} (x) = - \frac{80}{81 x^{\frac{11}{3}}}$