Calculus Examples

$f (x) = x^{\frac{9}{10}} - 6$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $x^{\frac{9}{10}} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{9}{10}}] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [x^{\frac{9}{10}}] + \frac{d}{d x} [- 6]$

Step 1.2

Evaluate $\frac{d}{d x} [x^{\frac{9}{10}}]$ .

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Step 1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{9}{10}$ .

$\frac{9}{10} x^{\frac{9}{10} - 1} + \frac{d}{d x} [- 6]$

Step 1.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{9}{10} x^{\frac{9}{10} - 1 \cdot \frac{10}{10}} + \frac{d}{d x} [- 6]$

Step 1.2.3

Combine $- 1$ and $\frac{10}{10}$ .

$\frac{9}{10} x^{\frac{9}{10} + \frac{- 1 \cdot 10}{10}} + \frac{d}{d x} [- 6]$

Step 1.2.4

Combine the numerators over the common denominator.

$\frac{9}{10} x^{\frac{9 - 1 \cdot 10}{10}} + \frac{d}{d x} [- 6]$

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $- 1$ by $10$ .

$\frac{9}{10} x^{\frac{9 - 10}{10}} + \frac{d}{d x} [- 6]$

Step 1.2.5.2

Subtract $10$ from $9$ .

$\frac{9}{10} x^{\frac{- 1}{10}} + \frac{d}{d x} [- 6]$

Step 1.2.6

Move the negative in front of the fraction.

$\frac{9}{10} x^{- \frac{1}{10}} + \frac{d}{d x} [- 6]$

Step 1.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{9}{10} x^{- \frac{1}{10}} + 0$

Step 1.4

Simplify.

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Step 1.4.1

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

$\frac{9}{10} \cdot \frac{1}{x^{\frac{1}{10}}} + 0$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Multiply $\frac{9}{10}$ by $\frac{1}{x^{\frac{1}{10}}}$ .

$\frac{9}{10 x^{\frac{1}{10}}} + 0$

Step 1.4.2.2

Add $\frac{9}{10 x^{\frac{1}{10}}}$ and $0$ .

$f' (x) = \frac{9}{10 x^{\frac{1}{10}}}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{9}{10}$ is constant with respect to $x$ , the derivative of $\frac{9}{10 x^{\frac{1}{10}}}$ with respect to $x$ is $\frac{9}{10} \frac{d}{d x} [\frac{1}{x^{\frac{1}{10}}}]$ .

$\frac{9}{10} \frac{d}{d x} [\frac{1}{x^{\frac{1}{10}}}]$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Rewrite $\frac{1}{x^{\frac{1}{10}}}$ as ${(x^{\frac{1}{10}})}^{- 1}$ .

$\frac{9}{10} \frac{d}{d x} [{(x^{\frac{1}{10}})}^{- 1}]$

Step 2.2.2

Multiply the exponents in ${(x^{\frac{1}{10}})}^{- 1}$ .

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Step 2.2.2.1

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

$\frac{9}{10} \frac{d}{d x} [x^{\frac{1}{10} \cdot - 1}]$

Step 2.2.2.2

Combine $\frac{1}{10}$ and $- 1$ .

$\frac{9}{10} \frac{d}{d x} [x^{\frac{- 1}{10}}]$

Step 2.2.2.3

Move the negative in front of the fraction.

$\frac{9}{10} \frac{d}{d x} [x^{- \frac{1}{10}}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{1}{10}$ .

$\frac{9}{10} (- \frac{1}{10} x^{- \frac{1}{10} - 1})$

Step 2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{9}{10} (- \frac{1}{10} x^{- \frac{1}{10} - 1 \cdot \frac{10}{10}})$

Step 2.5

Combine $- 1$ and $\frac{10}{10}$ .

$\frac{9}{10} (- \frac{1}{10} x^{- \frac{1}{10} + \frac{- 1 \cdot 10}{10}})$

Step 2.6

Combine the numerators over the common denominator.

$\frac{9}{10} (- \frac{1}{10} x^{\frac{- 1 - 1 \cdot 10}{10}})$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $10$ .

$\frac{9}{10} (- \frac{1}{10} x^{\frac{- 1 - 10}{10}})$

Step 2.7.2

Subtract $10$ from $- 1$ .

$\frac{9}{10} (- \frac{1}{10} x^{\frac{- 11}{10}})$

Step 2.8

Move the negative in front of the fraction.

$\frac{9}{10} (- \frac{1}{10} x^{- \frac{11}{10}})$

Step 2.9

Combine $x^{- \frac{11}{10}}$ and $\frac{1}{10}$ .

$\frac{9}{10} (- \frac{x^{- \frac{11}{10}}}{10})$

Step 2.10

Multiply $\frac{9}{10}$ by $\frac{x^{- \frac{11}{10}}}{10}$ .

$- \frac{9 x^{- \frac{11}{10}}}{10 \cdot 10}$

Step 2.11

Multiply.

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Step 2.11.1

Multiply $10$ by $10$ .

$- \frac{9 x^{- \frac{11}{10}}}{100}$

Step 2.11.2

Move $x^{- \frac{11}{10}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{9}{100 x^{\frac{11}{10}}}$

Step 3

Find the third derivative.

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Step 3.1

Since $- \frac{9}{100}$ is constant with respect to $x$ , the derivative of $- \frac{9}{100 x^{\frac{11}{10}}}$ with respect to $x$ is $- \frac{9}{100} \frac{d}{d x} [\frac{1}{x^{\frac{11}{10}}}]$ .

$- \frac{9}{100} \frac{d}{d x} [\frac{1}{x^{\frac{11}{10}}}]$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Rewrite $\frac{1}{x^{\frac{11}{10}}}$ as ${(x^{\frac{11}{10}})}^{- 1}$ .

$- \frac{9}{100} \frac{d}{d x} [{(x^{\frac{11}{10}})}^{- 1}]$

Step 3.2.2

Multiply the exponents in ${(x^{\frac{11}{10}})}^{- 1}$ .

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Step 3.2.2.1

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

$- \frac{9}{100} \frac{d}{d x} [x^{\frac{11}{10} \cdot - 1}]$

Step 3.2.2.2

Multiply $\frac{11}{10} \cdot - 1$ .

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Step 3.2.2.2.1

Combine $\frac{11}{10}$ and $- 1$ .

$- \frac{9}{100} \frac{d}{d x} [x^{\frac{11 \cdot - 1}{10}}]$

Step 3.2.2.2.2

Multiply $11$ by $- 1$ .

$- \frac{9}{100} \frac{d}{d x} [x^{\frac{- 11}{10}}]$

Step 3.2.2.3

Move the negative in front of the fraction.

$- \frac{9}{100} \frac{d}{d x} [x^{- \frac{11}{10}}]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{11}{10}$ .

$- \frac{9}{100} (- \frac{11}{10} x^{- \frac{11}{10} - 1})$

Step 3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$- \frac{9}{100} (- \frac{11}{10} x^{- \frac{11}{10} - 1 \cdot \frac{10}{10}})$

Step 3.5

Combine $- 1$ and $\frac{10}{10}$ .

$- \frac{9}{100} (- \frac{11}{10} x^{- \frac{11}{10} + \frac{- 1 \cdot 10}{10}})$

Step 3.6

Combine the numerators over the common denominator.

$- \frac{9}{100} (- \frac{11}{10} x^{\frac{- 11 - 1 \cdot 10}{10}})$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $10$ .

$- \frac{9}{100} (- \frac{11}{10} x^{\frac{- 11 - 10}{10}})$

Step 3.7.2

Subtract $10$ from $- 11$ .

$- \frac{9}{100} (- \frac{11}{10} x^{\frac{- 21}{10}})$

Step 3.8

Move the negative in front of the fraction.

$- \frac{9}{100} (- \frac{11}{10} x^{- \frac{21}{10}})$

Step 3.9

Combine $x^{- \frac{21}{10}}$ and $\frac{11}{10}$ .

$- \frac{9}{100} (- \frac{x^{- \frac{21}{10}} \cdot 11}{10})$

Step 3.10

Multiply.

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Step 3.10.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{9}{100}) \frac{x^{- \frac{21}{10}} \cdot 11}{10}$

Step 3.10.2

Multiply $\frac{9}{100}$ by $1$ .

$\frac{9}{100} \cdot \frac{x^{- \frac{21}{10}} \cdot 11}{10}$

Step 3.11

Multiply $\frac{9}{100}$ by $\frac{x^{- \frac{21}{10}} \cdot 11}{10}$ .

$\frac{9 (x^{- \frac{21}{10}} \cdot 11)}{100 \cdot 10}$

Step 3.12

Multiply.

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Step 3.12.1

Multiply $11$ by $9$ .

$\frac{99 x^{- \frac{21}{10}}}{100 \cdot 10}$

Step 3.12.2

Multiply $100$ by $10$ .

$\frac{99 x^{- \frac{21}{10}}}{1000}$

Step 3.12.3

Move $x^{- \frac{21}{10}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f''' (x) = \frac{99}{1000 x^{\frac{21}{10}}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $\frac{99}{1000}$ is constant with respect to $x$ , the derivative of $\frac{99}{1000 x^{\frac{21}{10}}}$ with respect to $x$ is $\frac{99}{1000} \frac{d}{d x} [\frac{1}{x^{\frac{21}{10}}}]$ .

$\frac{99}{1000} \frac{d}{d x} [\frac{1}{x^{\frac{21}{10}}}]$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

Rewrite $\frac{1}{x^{\frac{21}{10}}}$ as ${(x^{\frac{21}{10}})}^{- 1}$ .

$\frac{99}{1000} \frac{d}{d x} [{(x^{\frac{21}{10}})}^{- 1}]$

Step 4.2.2

Multiply the exponents in ${(x^{\frac{21}{10}})}^{- 1}$ .

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Step 4.2.2.1

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

$\frac{99}{1000} \frac{d}{d x} [x^{\frac{21}{10} \cdot - 1}]$

Step 4.2.2.2

Multiply $\frac{21}{10} \cdot - 1$ .

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Step 4.2.2.2.1

Combine $\frac{21}{10}$ and $- 1$ .

$\frac{99}{1000} \frac{d}{d x} [x^{\frac{21 \cdot - 1}{10}}]$

Step 4.2.2.2.2

Multiply $21$ by $- 1$ .

$\frac{99}{1000} \frac{d}{d x} [x^{\frac{- 21}{10}}]$

Step 4.2.2.3

Move the negative in front of the fraction.

$\frac{99}{1000} \frac{d}{d x} [x^{- \frac{21}{10}}]$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{21}{10}$ .

$\frac{99}{1000} (- \frac{21}{10} x^{- \frac{21}{10} - 1})$

Step 4.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{99}{1000} (- \frac{21}{10} x^{- \frac{21}{10} - 1 \cdot \frac{10}{10}})$

Step 4.5

Combine $- 1$ and $\frac{10}{10}$ .

$\frac{99}{1000} (- \frac{21}{10} x^{- \frac{21}{10} + \frac{- 1 \cdot 10}{10}})$

Step 4.6

Combine the numerators over the common denominator.

$\frac{99}{1000} (- \frac{21}{10} x^{\frac{- 21 - 1 \cdot 10}{10}})$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $10$ .

$\frac{99}{1000} (- \frac{21}{10} x^{\frac{- 21 - 10}{10}})$

Step 4.7.2

Subtract $10$ from $- 21$ .

$\frac{99}{1000} (- \frac{21}{10} x^{\frac{- 31}{10}})$

Step 4.8

Move the negative in front of the fraction.

$\frac{99}{1000} (- \frac{21}{10} x^{- \frac{31}{10}})$

Step 4.9

Combine $x^{- \frac{31}{10}}$ and $\frac{21}{10}$ .

$\frac{99}{1000} (- \frac{x^{- \frac{31}{10}} \cdot 21}{10})$

Step 4.10

Multiply $\frac{99}{1000}$ by $\frac{x^{- \frac{31}{10}} \cdot 21}{10}$ .

$- \frac{99 (x^{- \frac{31}{10}} \cdot 21)}{1000 \cdot 10}$

Step 4.11

Multiply.

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Step 4.11.1

Multiply $21$ by $99$ .

$- \frac{2079 x^{- \frac{31}{10}}}{1000 \cdot 10}$

Step 4.11.2

Multiply $1000$ by $10$ .

$- \frac{2079 x^{- \frac{31}{10}}}{10000}$

Step 4.11.3

Move $x^{- \frac{31}{10}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{4} (x) = - \frac{2079}{10000 x^{\frac{31}{10}}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $- \frac{2079}{10000 x^{\frac{31}{10}}}$ .

$- \frac{2079}{10000 x^{\frac{31}{10}}}$