Calculus Examples

$4 x - 5 x^{\frac{9}{10}}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $4 x - 5 x^{\frac{9}{10}}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 5 x^{\frac{9}{10}}]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- 5 x^{\frac{9}{10}}]$

Step 1.2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.2.1

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

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- 5 x^{\frac{9}{10}}]$

Step 1.2.2

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

$4 \cdot 1 + \frac{d}{d x} [- 5 x^{\frac{9}{10}}]$

Step 1.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- 5 x^{\frac{9}{10}}]$

Step 1.3

Evaluate $\frac{d}{d x} [- 5 x^{\frac{9}{10}}]$ .

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Step 1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{\frac{9}{10}}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{\frac{9}{10}}]$ .

$4 - 5 \frac{d}{d x} [x^{\frac{9}{10}}]$

Step 1.3.2

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

$4 - 5 (\frac{9}{10} x^{\frac{9}{10} - 1})$

Step 1.3.3

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

$4 - 5 (\frac{9}{10} x^{\frac{9}{10} - 1 \cdot \frac{10}{10}})$

Step 1.3.4

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

$4 - 5 (\frac{9}{10} x^{\frac{9}{10} + \frac{- 1 \cdot 10}{10}})$

Step 1.3.5

Combine the numerators over the common denominator.

$4 - 5 (\frac{9}{10} x^{\frac{9 - 1 \cdot 10}{10}})$

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Multiply $- 1$ by $10$ .

$4 - 5 (\frac{9}{10} x^{\frac{9 - 10}{10}})$

Step 1.3.6.2

Subtract $10$ from $9$ .

$4 - 5 (\frac{9}{10} x^{\frac{- 1}{10}})$

Step 1.3.7

Move the negative in front of the fraction.

$4 - 5 (\frac{9}{10} x^{- \frac{1}{10}})$

Step 1.3.8

Combine $\frac{9}{10}$ and $x^{- \frac{1}{10}}$ .

$4 - 5 \frac{9 x^{- \frac{1}{10}}}{10}$

Step 1.3.9

Combine $- 5$ and $\frac{9 x^{- \frac{1}{10}}}{10}$ .

$4 + \frac{- 5 (9 x^{- \frac{1}{10}})}{10}$

Step 1.3.10

Multiply $9$ by $- 5$ .

$4 + \frac{- 45 x^{- \frac{1}{10}}}{10}$

Step 1.3.11

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

$4 + \frac{- 45}{10 x^{\frac{1}{10}}}$

Step 1.3.12

Factor $5$ out of $- 45$ .

$4 + \frac{5 (- 9)}{10 x^{\frac{1}{10}}}$

Step 1.3.13

Cancel the common factors.

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Step 1.3.13.1

Factor $5$ out of $10 x^{\frac{1}{10}}$ .

$4 + \frac{5 (- 9)}{5 (2 x^{\frac{1}{10}})}$

Step 1.3.13.2

Cancel the common factor.

$4 + \frac{5 \cdot - 9}{5 (2 x^{\frac{1}{10}})}$

Step 1.3.13.3

Rewrite the expression.

$4 + \frac{- 9}{2 x^{\frac{1}{10}}}$

Step 1.3.14

Move the negative in front of the fraction.

$f' (x) = 4 - \frac{9}{2 x^{\frac{1}{10}}}$

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $4 - \frac{9}{2 x^{\frac{1}{10}}}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- \frac{9}{2 x^{\frac{1}{10}}}]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- \frac{9}{2 x^{\frac{1}{10}}}]$

Step 2.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{9}{2 x^{\frac{1}{10}}}]$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{9}{2 x^{\frac{1}{10}}}]$ .

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Step 2.2.1

Since $- \frac{9}{2}$ is constant with respect to $x$ , the derivative of $- \frac{9}{2 x^{\frac{1}{10}}}$ with respect to $x$ is $- \frac{9}{2} \frac{d}{d x} [\frac{1}{x^{\frac{1}{10}}}]$ .

$0 - \frac{9}{2} \frac{d}{d x} [\frac{1}{x^{\frac{1}{10}}}]$

Step 2.2.2

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

$0 - \frac{9}{2} \frac{d}{d x} [{(x^{\frac{1}{10}})}^{- 1}]$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{1}{10}}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{10}}$ .

$0 - \frac{9}{2} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{1}{10}}])$

Step 2.2.3.2

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

$0 - \frac{9}{2} (- u^{- 2} \frac{d}{d x} [x^{\frac{1}{10}}])$

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{10}}$ .

$0 - \frac{9}{2} (- {(x^{\frac{1}{10}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{10}}])$

Step 2.2.4

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

$0 - \frac{9}{2} (- {(x^{\frac{1}{10}})}^{- 2} (\frac{1}{10} x^{\frac{1}{10} - 1}))$

Step 2.2.5

Multiply the exponents in ${(x^{\frac{1}{10}})}^{- 2}$ .

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Step 2.2.5.1

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

$0 - \frac{9}{2} (- x^{\frac{1}{10} \cdot - 2} (\frac{1}{10} x^{\frac{1}{10} - 1}))$

Step 2.2.5.2

Cancel the common factor of $2$ .

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Step 2.2.5.2.1

Factor $2$ out of $10$ .

$0 - \frac{9}{2} (- x^{\frac{1}{2 (5)} \cdot - 2} (\frac{1}{10} x^{\frac{1}{10} - 1}))$

Step 2.2.5.2.2

Factor $2$ out of $- 2$ .

$0 - \frac{9}{2} (- x^{\frac{1}{2 \cdot 5} \cdot (2 \cdot - 1)} (\frac{1}{10} x^{\frac{1}{10} - 1}))$

Step 2.2.5.2.3

Cancel the common factor.

$0 - \frac{9}{2} (- x^{\frac{1}{2 \cdot 5} \cdot (2 \cdot - 1)} (\frac{1}{10} x^{\frac{1}{10} - 1}))$

Step 2.2.5.2.4

Rewrite the expression.

$0 - \frac{9}{2} (- x^{\frac{1}{5} \cdot - 1} (\frac{1}{10} x^{\frac{1}{10} - 1}))$

Step 2.2.5.3

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

$0 - \frac{9}{2} (- x^{\frac{- 1}{5}} (\frac{1}{10} x^{\frac{1}{10} - 1}))$

Step 2.2.5.4

Move the negative in front of the fraction.

$0 - \frac{9}{2} (- x^{- \frac{1}{5}} (\frac{1}{10} x^{\frac{1}{10} - 1}))$

Step 2.2.6

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

$0 - \frac{9}{2} (- x^{- \frac{1}{5}} (\frac{1}{10} x^{\frac{1}{10} - 1 \cdot \frac{10}{10}}))$

Step 2.2.7

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

$0 - \frac{9}{2} (- x^{- \frac{1}{5}} (\frac{1}{10} x^{\frac{1}{10} + \frac{- 1 \cdot 10}{10}}))$

Step 2.2.8

Combine the numerators over the common denominator.

$0 - \frac{9}{2} (- x^{- \frac{1}{5}} (\frac{1}{10} x^{\frac{1 - 1 \cdot 10}{10}}))$

Step 2.2.9

Simplify the numerator.

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Step 2.2.9.1

Multiply $- 1$ by $10$ .

$0 - \frac{9}{2} (- x^{- \frac{1}{5}} (\frac{1}{10} x^{\frac{1 - 10}{10}}))$

Step 2.2.9.2

Subtract $10$ from $1$ .

$0 - \frac{9}{2} (- x^{- \frac{1}{5}} (\frac{1}{10} x^{\frac{- 9}{10}}))$

Step 2.2.10

Move the negative in front of the fraction.

$0 - \frac{9}{2} (- x^{- \frac{1}{5}} (\frac{1}{10} x^{- \frac{9}{10}}))$

Step 2.2.11

Combine $\frac{1}{10}$ and $x^{- \frac{9}{10}}$ .

$0 - \frac{9}{2} (- x^{- \frac{1}{5}} \frac{x^{- \frac{9}{10}}}{10})$

Step 2.2.12

Combine $\frac{x^{- \frac{9}{10}}}{10}$ and $x^{- \frac{1}{5}}$ .

$0 - \frac{9}{2} (- \frac{x^{- \frac{9}{10}} x^{- \frac{1}{5}}}{10})$

Step 2.2.13

Multiply $x^{- \frac{9}{10}}$ by $x^{- \frac{1}{5}}$ by adding the exponents.

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Step 2.2.13.1

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

$0 - \frac{9}{2} (- \frac{x^{- \frac{9}{10} - \frac{1}{5}}}{10})$

Step 2.2.13.2

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$0 - \frac{9}{2} (- \frac{x^{- \frac{9}{10} - \frac{1}{5} \cdot \frac{2}{2}}}{10})$

Step 2.2.13.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.13.3.1

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

$0 - \frac{9}{2} (- \frac{x^{- \frac{9}{10} - \frac{2}{5 \cdot 2}}}{10})$

Step 2.2.13.3.2

Multiply $5$ by $2$ .

$0 - \frac{9}{2} (- \frac{x^{- \frac{9}{10} - \frac{2}{10}}}{10})$

Step 2.2.13.4

Combine the numerators over the common denominator.

$0 - \frac{9}{2} (- \frac{x^{\frac{- 9 - 2}{10}}}{10})$

Step 2.2.13.5

Subtract $2$ from $- 9$ .

$0 - \frac{9}{2} (- \frac{x^{\frac{- 11}{10}}}{10})$

Step 2.2.13.6

Move the negative in front of the fraction.

$0 - \frac{9}{2} (- \frac{x^{- \frac{11}{10}}}{10})$

Step 2.2.14

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

$0 - \frac{9}{2} (- \frac{1}{10 x^{\frac{11}{10}}})$

Step 2.2.15

Multiply $- 1$ by $- 1$ .

$0 + 1 (\frac{9}{2}) \frac{1}{10 x^{\frac{11}{10}}}$

Step 2.2.16

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

$0 + \frac{9}{2} \cdot \frac{1}{10 x^{\frac{11}{10}}}$

Step 2.2.17

Multiply $\frac{9}{2}$ by $\frac{1}{10 x^{\frac{11}{10}}}$ .

$0 + \frac{9}{2 (10 x^{\frac{11}{10}})}$

Step 2.2.18

Multiply $10$ by $2$ .

$0 + \frac{9}{20 x^{\frac{11}{10}}}$

Step 2.3

Add $0$ and $\frac{9}{20 x^{\frac{11}{10}}}$ .

$f'' (x) = \frac{9}{20 x^{\frac{11}{10}}}$