Calculus Examples

$y = \sqrt[5]{x}$

Step 1

Find the first derivative.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$f' (x) = \frac{d}{d x} (x^{\frac{1}{5}})$

Step 1.2

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

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

Step 1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $5$ .

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

Step 1.6.2

Subtract $5$ from $1$ .

$f' (x) = \frac{1}{5} x^{\frac{- 4}{5}}$

Step 1.7

Move the negative in front of the fraction.

$f' (x) = \frac{1}{5} x^{- \frac{4}{5}}$

Step 1.8

Simplify.

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Step 1.8.1

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

$f' (x) = \frac{1}{5} \cdot \frac{1}{x^{\frac{4}{5}}}$

Step 1.8.2

Multiply $\frac{1}{5}$ by $\frac{1}{x^{\frac{4}{5}}}$ .

$f' (x) = \frac{1}{5 x^{\frac{4}{5}}}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{1}{5 x^{\frac{4}{5}}}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [\frac{1}{x^{\frac{4}{5}}}]$ .

$f'' (x) = \frac{1}{5} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{4}{5}}})$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Rewrite $\frac{1}{x^{\frac{4}{5}}}$ as ${(x^{\frac{4}{5}})}^{- 1}$ .

$f'' (x) = \frac{1}{5} \cdot \frac{d}{d x} ({(x^{\frac{4}{5}})}^{- 1})$

Step 2.2.2

Multiply the exponents in ${(x^{\frac{4}{5}})}^{- 1}$ .

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Step 2.2.2.1

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

$f'' (x) = \frac{1}{5} \cdot \frac{d}{d x} (x^{\frac{4}{5} \cdot - 1})$

Step 2.2.2.2

Multiply $\frac{4}{5} \cdot - 1$ .

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Step 2.2.2.2.1

Combine $\frac{4}{5}$ and $- 1$ .

$f'' (x) = \frac{1}{5} \cdot \frac{d}{d x} (x^{\frac{4 \cdot - 1}{5}})$

Step 2.2.2.2.2

Multiply $4$ by $- 1$ .

$f'' (x) = \frac{1}{5} \cdot \frac{d}{d x} (x^{\frac{- 4}{5}})$

Step 2.2.2.3

Move the negative in front of the fraction.

$f'' (x) = \frac{1}{5} \cdot \frac{d}{d x} (x^{- \frac{4}{5}})$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{4}{5}$ .

$f'' (x) = \frac{1}{5} \cdot (- \frac{4}{5} x^{- \frac{4}{5} - 1})$

Step 2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f'' (x) = \frac{1}{5} \cdot (- \frac{4}{5} x^{- \frac{4}{5} - 1 \cdot \frac{5}{5}})$

Step 2.5

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

$f'' (x) = \frac{1}{5} \cdot (- \frac{4}{5} x^{- \frac{4}{5} + \frac{- 1 \cdot 5}{5}})$

Step 2.6

Combine the numerators over the common denominator.

$f'' (x) = \frac{1}{5} \cdot (- \frac{4}{5} x^{\frac{- 4 - 1 \cdot 5}{5}})$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $5$ .

$f'' (x) = \frac{1}{5} \cdot (- \frac{4}{5} x^{\frac{- 4 - 5}{5}})$

Step 2.7.2

Subtract $5$ from $- 4$ .

$f'' (x) = \frac{1}{5} \cdot (- \frac{4}{5} x^{\frac{- 9}{5}})$

Step 2.8

Move the negative in front of the fraction.

$f'' (x) = \frac{1}{5} \cdot (- \frac{4}{5} x^{- \frac{9}{5}})$

Step 2.9

Combine $x^{- \frac{9}{5}}$ and $\frac{4}{5}$ .

$f'' (x) = \frac{1}{5} \cdot (- \frac{x^{- \frac{9}{5}} \cdot 4}{5})$

Step 2.10

Multiply $\frac{1}{5}$ by $\frac{x^{- \frac{9}{5}} \cdot 4}{5}$ .

$f'' (x) = - \frac{x^{- \frac{9}{5}} \cdot 4}{5 \cdot 5}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Multiply $5$ by $5$ .

$f'' (x) = - \frac{x^{- \frac{9}{5}} \cdot 4}{25}$

Step 2.11.2

Move $4$ to the left of $x^{- \frac{9}{5}}$ .

$f'' (x) = - \frac{4 \cdot x^{- \frac{9}{5}}}{25}$

Step 2.11.3

Move $x^{- \frac{9}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{4}{25 x^{\frac{9}{5}}}$

Step 3

Find the third derivative.

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Step 3.1

Since $- \frac{4}{25}$ is constant with respect to $x$ , the derivative of $- \frac{4}{25 x^{\frac{9}{5}}}$ with respect to $x$ is $- \frac{4}{25} \frac{d}{d x} [\frac{1}{x^{\frac{9}{5}}}]$ .

$f''' (x) = - \frac{4}{25} \cdot \frac{d}{d x} \frac{1}{x^{\frac{9}{5}}}$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Rewrite $\frac{1}{x^{\frac{9}{5}}}$ as ${(x^{\frac{9}{5}})}^{- 1}$ .

$f''' (x) = - \frac{4}{25} \cdot \frac{d}{d x} {(x^{\frac{9}{5}})}^{- 1}$

Step 3.2.2

Multiply the exponents in ${(x^{\frac{9}{5}})}^{- 1}$ .

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Step 3.2.2.1

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

$f''' (x) = - \frac{4}{25} \cdot \frac{d}{d x} x^{\frac{9}{5} \cdot - 1}$

Step 3.2.2.2

Multiply $\frac{9}{5} \cdot - 1$ .

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Step 3.2.2.2.1

Combine $\frac{9}{5}$ and $- 1$ .

$f''' (x) = - \frac{4}{25} \cdot \frac{d}{d x} x^{\frac{9 \cdot - 1}{5}}$

Step 3.2.2.2.2

Multiply $9$ by $- 1$ .

$f''' (x) = - \frac{4}{25} \cdot \frac{d}{d x} x^{\frac{- 9}{5}}$

Step 3.2.2.3

Move the negative in front of the fraction.

$f''' (x) = - \frac{4}{25} \cdot \frac{d}{d x} x^{- \frac{9}{5}}$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{9}{5}$ .

$f''' (x) = - \frac{4}{25} \cdot (- \frac{9}{5} x^{- \frac{9}{5} - 1})$

Step 3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f''' (x) = - \frac{4}{25} \cdot (- \frac{9}{5} x^{- \frac{9}{5} - 1 \cdot \frac{5}{5}})$

Step 3.5

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

$f''' (x) = - \frac{4}{25} \cdot (- \frac{9}{5} x^{- \frac{9}{5} + \frac{- 1 \cdot 5}{5}})$

Step 3.6

Combine the numerators over the common denominator.

$f''' (x) = - \frac{4}{25} \cdot (- \frac{9}{5} x^{\frac{- 9 - 1 \cdot 5}{5}})$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $5$ .

$f''' (x) = - \frac{4}{25} \cdot (- \frac{9}{5} x^{\frac{- 9 - 5}{5}})$

Step 3.7.2

Subtract $5$ from $- 9$ .

$f''' (x) = - \frac{4}{25} \cdot (- \frac{9}{5} x^{\frac{- 14}{5}})$

Step 3.8

Move the negative in front of the fraction.

$f''' (x) = - \frac{4}{25} \cdot (- \frac{9}{5} x^{- \frac{14}{5}})$

Step 3.9

Combine $x^{- \frac{14}{5}}$ and $\frac{9}{5}$ .

$f''' (x) = - \frac{4}{25} \cdot (- \frac{x^{- \frac{14}{5}} \cdot 9}{5})$

Step 3.10

Multiply.

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Step 3.10.1

Multiply $- 1$ by $- 1$ .

$f''' (x) = 1 (\frac{4}{25}) \cdot \frac{x^{- \frac{14}{5}} \cdot 9}{5}$

Step 3.10.2

Multiply $\frac{4}{25}$ by $1$ .

$f''' (x) = \frac{4}{25} \cdot \frac{x^{- \frac{14}{5}} \cdot 9}{5}$

Step 3.11

Multiply $\frac{4}{25}$ by $\frac{x^{- \frac{14}{5}} \cdot 9}{5}$ .

$f''' (x) = \frac{4 (x^{- \frac{14}{5}} \cdot 9)}{25 \cdot 5}$

Step 3.12

Multiply.

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Step 3.12.1

Multiply $9$ by $4$ .

$f''' (x) = \frac{36 x^{- \frac{14}{5}}}{25 \cdot 5}$

Step 3.12.2

Multiply $25$ by $5$ .

$f''' (x) = \frac{36 x^{- \frac{14}{5}}}{125}$

Step 3.12.3

Move $x^{- \frac{14}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f''' (x) = \frac{36}{125 x^{\frac{14}{5}}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $\frac{36}{125}$ is constant with respect to $x$ , the derivative of $\frac{36}{125 x^{\frac{14}{5}}}$ with respect to $x$ is $\frac{36}{125} \frac{d}{d x} [\frac{1}{x^{\frac{14}{5}}}]$ .

$f^{4} (x) = \frac{36}{125} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{14}{5}}})$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

Rewrite $\frac{1}{x^{\frac{14}{5}}}$ as ${(x^{\frac{14}{5}})}^{- 1}$ .

$f^{4} (x) = \frac{36}{125} \cdot \frac{d}{d x} ({(x^{\frac{14}{5}})}^{- 1})$

Step 4.2.2

Multiply the exponents in ${(x^{\frac{14}{5}})}^{- 1}$ .

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Step 4.2.2.1

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

$f^{4} (x) = \frac{36}{125} \cdot \frac{d}{d x} (x^{\frac{14}{5} \cdot - 1})$

Step 4.2.2.2

Multiply $\frac{14}{5} \cdot - 1$ .

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Step 4.2.2.2.1

Combine $\frac{14}{5}$ and $- 1$ .

$f^{4} (x) = \frac{36}{125} \cdot \frac{d}{d x} (x^{\frac{14 \cdot - 1}{5}})$

Step 4.2.2.2.2

Multiply $14$ by $- 1$ .

$f^{4} (x) = \frac{36}{125} \cdot \frac{d}{d x} (x^{\frac{- 14}{5}})$

Step 4.2.2.3

Move the negative in front of the fraction.

$f^{4} (x) = \frac{36}{125} \cdot \frac{d}{d x} (x^{- \frac{14}{5}})$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{14}{5}$ .

$f^{4} (x) = \frac{36}{125} \cdot (- \frac{14}{5} x^{- \frac{14}{5} - 1})$

Step 4.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f^{4} (x) = \frac{36}{125} \cdot (- \frac{14}{5} x^{- \frac{14}{5} - 1 \cdot \frac{5}{5}})$

Step 4.5

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

$f^{4} (x) = \frac{36}{125} \cdot (- \frac{14}{5} x^{- \frac{14}{5} + \frac{- 1 \cdot 5}{5}})$

Step 4.6

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{36}{125} \cdot (- \frac{14}{5} x^{\frac{- 14 - 1 \cdot 5}{5}})$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $5$ .

$f^{4} (x) = \frac{36}{125} \cdot (- \frac{14}{5} x^{\frac{- 14 - 5}{5}})$

Step 4.7.2

Subtract $5$ from $- 14$ .

$f^{4} (x) = \frac{36}{125} \cdot (- \frac{14}{5} x^{\frac{- 19}{5}})$

Step 4.8

Move the negative in front of the fraction.

$f^{4} (x) = \frac{36}{125} \cdot (- \frac{14}{5} x^{- \frac{19}{5}})$

Step 4.9

Combine $x^{- \frac{19}{5}}$ and $\frac{14}{5}$ .

$f^{4} (x) = \frac{36}{125} \cdot (- \frac{x^{- \frac{19}{5}} \cdot 14}{5})$

Step 4.10

Multiply $\frac{36}{125}$ by $\frac{x^{- \frac{19}{5}} \cdot 14}{5}$ .

$f^{4} (x) = - \frac{36 (x^{- \frac{19}{5}} \cdot 14)}{125 \cdot 5}$

Step 4.11

Multiply.

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Step 4.11.1

Multiply $14$ by $36$ .

$f^{4} (x) = - \frac{504 x^{- \frac{19}{5}}}{125 \cdot 5}$

Step 4.11.2

Multiply $125$ by $5$ .

$f^{4} (x) = - \frac{504 x^{- \frac{19}{5}}}{625}$

Step 4.11.3

Move $x^{- \frac{19}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{4} (x) = - \frac{504}{625 x^{\frac{19}{5}}}$