Calculus Examples

$f (x) = \sqrt{x} + \sqrt[5]{x}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\sqrt{x} + \sqrt[5]{x}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [\sqrt[5]{x}]$ .

$f' (x) = \frac{d}{d x} (\sqrt{x}) + \frac{d}{d x} (\sqrt[5]{x})$

Step 1.2

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

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$f' (x) = \frac{d}{d x} (x^{\frac{1}{2}}) + \frac{d}{d x} (\sqrt[5]{x})$

Step 1.2.2

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

$f' (x) = \frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} (\sqrt[5]{x})$

Step 1.2.3

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

$f' (x) = \frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} (\sqrt[5]{x})$

Step 1.2.4

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

$f' (x) = \frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} (\sqrt[5]{x})$

Step 1.2.5

Combine the numerators over the common denominator.

$f' (x) = \frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} (\sqrt[5]{x})$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $2$ .

$f' (x) = \frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} (\sqrt[5]{x})$

Step 1.2.6.2

Subtract $2$ from $1$ .

$f' (x) = \frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} (\sqrt[5]{x})$

Step 1.2.7

Move the negative in front of the fraction.

$f' (x) = \frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} (\sqrt[5]{x})$

Step 1.3

Evaluate $\frac{d}{d x} [\sqrt[5]{x}]$ .

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Combine the numerators over the common denominator.

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

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Multiply $- 1$ by $5$ .

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

Step 1.3.6.2

Subtract $5$ from $1$ .

$f' (x) = \frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{5} x^{\frac{- 4}{5}}$

Step 1.3.7

Move the negative in front of the fraction.

$f' (x) = \frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{5} x^{- \frac{4}{5}}$

Step 1.4

Simplify.

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Step 1.4.1

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

$f' (x) = \frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} + \frac{1}{5} x^{- \frac{4}{5}}$

Step 1.4.2

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

$f' (x) = \frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} + \frac{1}{5} \cdot \frac{1}{x^{\frac{4}{5}}}$

Step 1.4.3

Combine terms.

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Step 1.4.3.1

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

$f' (x) = \frac{1}{2 x^{\frac{1}{2}}} + \frac{1}{5} \cdot \frac{1}{x^{\frac{4}{5}}}$

Step 1.4.3.2

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

$f' (x) = \frac{1}{2 x^{\frac{1}{2}}} + \frac{1}{5 x^{\frac{4}{5}}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{1}{2 x^{\frac{1}{2}}} + \frac{1}{5 x^{\frac{4}{5}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{2 x^{\frac{1}{2}}}] + \frac{d}{d x} [\frac{1}{5 x^{\frac{4}{5}}}]$ .

$f'' (x) = \frac{d}{d x} (\frac{1}{2 x^{\frac{1}{2}}}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = \frac{1}{2} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{1}{2}}}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.2

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

$f'' (x) = \frac{1}{2} \cdot \frac{d}{d x} ({(x^{\frac{1}{2}})}^{- 1}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{1}{2}}$ .

$f'' (x) = \frac{1}{2} \cdot (\frac{d}{d u (1)} ({u_{1}}^{- 1}) \frac{d}{d x} (x^{\frac{1}{2}})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.3.2

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

$f'' (x) = \frac{1}{2} \cdot (- {u_{1}}^{- 2} \frac{d}{d x} x^{\frac{1}{2}}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $x^{\frac{1}{2}}$ .

$f'' (x) = \frac{1}{2} \cdot (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} x^{\frac{1}{2}}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.4

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

$f'' (x) = \frac{1}{2} \cdot (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.5

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

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Step 2.2.5.1

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

$f'' (x) = \frac{1}{2} \cdot (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

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 $- 2$ .

$f'' (x) = \frac{1}{2} \cdot (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.5.2.2

Cancel the common factor.

$f'' (x) = \frac{1}{2} \cdot (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.5.2.3

Rewrite the expression.

$f'' (x) = \frac{1}{2} \cdot (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.6

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

$f'' (x) = \frac{1}{2} \cdot (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.7

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

$f'' (x) = \frac{1}{2} \cdot (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.8

Combine the numerators over the common denominator.

$f'' (x) = \frac{1}{2} \cdot (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.9

Simplify the numerator.

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Step 2.2.9.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{1}{2} \cdot (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.9.2

Subtract $2$ from $1$ .

$f'' (x) = \frac{1}{2} \cdot (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.10

Move the negative in front of the fraction.

$f'' (x) = \frac{1}{2} \cdot (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}})) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.11

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$f'' (x) = \frac{1}{2} \cdot (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.12

Combine $\frac{x^{- \frac{1}{2}}}{2}$ and $x^{- 1}$ .

$f'' (x) = \frac{1}{2} \cdot (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.13

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

$f'' (x) = \frac{1}{2} \cdot (- \frac{x^{- \frac{1}{2} - 1}}{2}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.13.2

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

$f'' (x) = \frac{1}{2} \cdot (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.13.3

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

$f'' (x) = \frac{1}{2} \cdot (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.13.4

Combine the numerators over the common denominator.

$f'' (x) = \frac{1}{2} \cdot (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.13.5

Simplify the numerator.

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Step 2.2.13.5.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{1}{2} \cdot (- \frac{x^{\frac{- 1 - 2}{2}}}{2}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.13.5.2

Subtract $2$ from $- 1$ .

$f'' (x) = \frac{1}{2} \cdot (- \frac{x^{\frac{- 3}{2}}}{2}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.13.6

Move the negative in front of the fraction.

$f'' (x) = \frac{1}{2} \cdot (- \frac{x^{- \frac{3}{2}}}{2}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.14

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{1}{2} \cdot (- \frac{1}{2 x^{\frac{3}{2}}}) + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.15

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

$f'' (x) = - \frac{1}{2 (2 x^{\frac{3}{2}})} + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.2.16

Multiply $2$ by $2$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{d}{d x} (\frac{1}{5 x^{\frac{4}{5}}})$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{1}{5 x^{\frac{4}{5}}}]$ .

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Step 2.3.1

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 $\frac{1}{5} \frac{d}{d x} [\frac{1}{x^{\frac{4}{5}}}]$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{4}{5}}})$

Step 2.3.2

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

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot \frac{d}{d x} ({(x^{\frac{4}{5}})}^{- 1})$

Step 2.3.3

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 $g (x) = x^{\frac{4}{5}}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{4}{5}}$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (\frac{d}{d u (2)} ({u_{2}}^{- 1}) \frac{d}{d x} (x^{\frac{4}{5}}))$

Step 2.3.3.2

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

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- {u_{2}}^{- 2} \frac{d}{d x} x^{\frac{4}{5}})$

Step 2.3.3.3

Replace all occurrences of $u_{2}$ with $x^{\frac{4}{5}}$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- {(x^{\frac{4}{5}})}^{- 2} \frac{d}{d x} x^{\frac{4}{5}})$

Step 2.3.4

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

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- {(x^{\frac{4}{5}})}^{- 2} (\frac{4}{5} x^{\frac{4}{5} - 1}))$

Step 2.3.5

Multiply the exponents in ${(x^{\frac{4}{5}})}^{- 2}$ .

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Step 2.3.5.1

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

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{\frac{4}{5} \cdot - 2} (\frac{4}{5} x^{\frac{4}{5} - 1}))$

Step 2.3.5.2

Multiply $\frac{4}{5} \cdot - 2$ .

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Step 2.3.5.2.1

Combine $\frac{4}{5}$ and $- 2$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{\frac{4 \cdot - 2}{5}} (\frac{4}{5} x^{\frac{4}{5} - 1}))$

Step 2.3.5.2.2

Multiply $4$ by $- 2$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{\frac{- 8}{5}} (\frac{4}{5} x^{\frac{4}{5} - 1}))$

Step 2.3.5.3

Move the negative in front of the fraction.

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{- \frac{8}{5}} (\frac{4}{5} x^{\frac{4}{5} - 1}))$

Step 2.3.6

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

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{- \frac{8}{5}} (\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}}))$

Step 2.3.7

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

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{- \frac{8}{5}} (\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}}))$

Step 2.3.8

Combine the numerators over the common denominator.

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{- \frac{8}{5}} (\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}}))$

Step 2.3.9

Simplify the numerator.

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Step 2.3.9.1

Multiply $- 1$ by $5$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{- \frac{8}{5}} (\frac{4}{5} x^{\frac{4 - 5}{5}}))$

Step 2.3.9.2

Subtract $5$ from $4$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{- \frac{8}{5}} (\frac{4}{5} x^{\frac{- 1}{5}}))$

Step 2.3.10

Move the negative in front of the fraction.

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{- \frac{8}{5}} (\frac{4}{5} x^{- \frac{1}{5}}))$

Step 2.3.11

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

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- x^{- \frac{8}{5}} \frac{4 x^{- \frac{1}{5}}}{5})$

Step 2.3.12

Combine $\frac{4 x^{- \frac{1}{5}}}{5}$ and $x^{- \frac{8}{5}}$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- \frac{4 x^{- \frac{1}{5}} x^{- \frac{8}{5}}}{5})$

Step 2.3.13

Multiply $x^{- \frac{1}{5}}$ by $x^{- \frac{8}{5}}$ by adding the exponents.

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Step 2.3.13.1

Move $x^{- \frac{8}{5}}$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- \frac{4 (x^{- \frac{8}{5}} x^{- \frac{1}{5}})}{5})$

Step 2.3.13.2

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

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- \frac{4 x^{- \frac{8}{5} - \frac{1}{5}}}{5})$

Step 2.3.13.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- \frac{4 x^{\frac{- 8 - 1}{5}}}{5})$

Step 2.3.13.4

Subtract $1$ from $- 8$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- \frac{4 x^{\frac{- 9}{5}}}{5})$

Step 2.3.13.5

Move the negative in front of the fraction.

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- \frac{4 x^{- \frac{9}{5}}}{5})$

Step 2.3.14

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

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} + \frac{1}{5} \cdot (- \frac{4}{5 x^{\frac{9}{5}}})$

Step 2.3.15

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

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} - \frac{4}{5 (5 x^{\frac{9}{5}})}$

Step 2.3.16

Multiply $5$ by $5$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}} - \frac{4}{25 x^{\frac{9}{5}}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{1}{4 x^{\frac{3}{2}}} - \frac{4}{25 x^{\frac{9}{5}}}$ .

$- \frac{1}{4 x^{\frac{3}{2}}} - \frac{4}{25 x^{\frac{9}{5}}}$