Calculus Examples

$f (x) = x^{4} + 3 x^{2} - \frac{2}{\sqrt{x}}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{4} + 3 x^{2} - \frac{2}{\sqrt{x}}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- \frac{2}{\sqrt{x}}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- \frac{2}{\sqrt{x}}]$

Step 1.1.2

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

$4 x^{3} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- \frac{2}{\sqrt{x}}]$

Step 1.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 1.2.1

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

$4 x^{3} + 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{2}{\sqrt{x}}]$

Step 1.2.2

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

$4 x^{3} + 3 (2 x) + \frac{d}{d x} [- \frac{2}{\sqrt{x}}]$

Step 1.2.3

Multiply $2$ by $3$ .

$4 x^{3} + 6 x + \frac{d}{d x} [- \frac{2}{\sqrt{x}}]$

Step 1.3

Evaluate $\frac{d}{d x} [- \frac{2}{\sqrt{x}}]$ .

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Step 1.3.1

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

$4 x^{3} + 6 x + \frac{d}{d x} [- \frac{2}{x^{\frac{1}{2}}}]$

Step 1.3.2

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

$4 x^{3} + 6 x - 2 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$

Step 1.3.3

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

$4 x^{3} + 6 x - 2 \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}]$

Step 1.3.4

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 1.3.4.1

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

$4 x^{3} + 6 x - 2 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 1.3.4.2

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

$4 x^{3} + 6 x - 2 (- u^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 1.3.4.3

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

$4 x^{3} + 6 x - 2 (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 1.3.5

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

$4 x^{3} + 6 x - 2 (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 1.3.6

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

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Step 1.3.6.1

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

$4 x^{3} + 6 x - 2 (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 1.3.6.2

Cancel the common factor of $2$ .

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Step 1.3.6.2.1

Factor $2$ out of $- 2$ .

$4 x^{3} + 6 x - 2 (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 1.3.6.2.2

Cancel the common factor.

$4 x^{3} + 6 x - 2 (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 1.3.6.2.3

Rewrite the expression.

$4 x^{3} + 6 x - 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 1.3.7

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

$4 x^{3} + 6 x - 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 1.3.8

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

$4 x^{3} + 6 x - 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 1.3.9

Combine the numerators over the common denominator.

$4 x^{3} + 6 x - 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 1.3.10

Simplify the numerator.

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Step 1.3.10.1

Multiply $- 1$ by $2$ .

$4 x^{3} + 6 x - 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 1.3.10.2

Subtract $2$ from $1$ .

$4 x^{3} + 6 x - 2 (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 1.3.11

Move the negative in front of the fraction.

$4 x^{3} + 6 x - 2 (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 1.3.12

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

$4 x^{3} + 6 x - 2 (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2})$

Step 1.3.13

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

$4 x^{3} + 6 x - 2 (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2})$

Step 1.3.14

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

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Step 1.3.14.1

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

$4 x^{3} + 6 x - 2 (- \frac{x^{- \frac{1}{2} - 1}}{2})$

Step 1.3.14.2

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

$4 x^{3} + 6 x - 2 (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2})$

Step 1.3.14.3

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

$4 x^{3} + 6 x - 2 (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2})$

Step 1.3.14.4

Combine the numerators over the common denominator.

$4 x^{3} + 6 x - 2 (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2})$

Step 1.3.14.5

Simplify the numerator.

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Step 1.3.14.5.1

Multiply $- 1$ by $2$ .

$4 x^{3} + 6 x - 2 (- \frac{x^{\frac{- 1 - 2}{2}}}{2})$

Step 1.3.14.5.2

Subtract $2$ from $- 1$ .

$4 x^{3} + 6 x - 2 (- \frac{x^{\frac{- 3}{2}}}{2})$

Step 1.3.14.6

Move the negative in front of the fraction.

$4 x^{3} + 6 x - 2 (- \frac{x^{- \frac{3}{2}}}{2})$

Step 1.3.15

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

$4 x^{3} + 6 x - 2 (- \frac{1}{2 x^{\frac{3}{2}}})$

Step 1.3.16

Multiply $- 1$ by $- 2$ .

$4 x^{3} + 6 x + 2 \frac{1}{2 x^{\frac{3}{2}}}$

Step 1.3.17

Combine $2$ and $\frac{1}{2 x^{\frac{3}{2}}}$ .

$4 x^{3} + 6 x + \frac{2}{2 x^{\frac{3}{2}}}$

Step 1.3.18

Cancel the common factor.

$4 x^{3} + 6 x + \frac{2}{2 x^{\frac{3}{2}}}$

Step 1.3.19

Rewrite the expression.

$f' (x) = 4 x^{3} + 6 x + \frac{1}{x^{\frac{3}{2}}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $4 x^{3} + 6 x + \frac{1}{x^{\frac{3}{2}}}$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$

Step 2.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$

Step 2.2.2

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

$4 (3 x^{2}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$

Step 2.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [6 x] + \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$

Step 2.3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$12 x^{2} + 6 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$

Step 2.3.2

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

$12 x^{2} + 6 \cdot 1 + \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$

Step 2.3.3

Multiply $6$ by $1$ .

$12 x^{2} + 6 + \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$

Step 2.4

Evaluate $\frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$ .

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Step 2.4.1

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

$12 x^{2} + 6 + \frac{d}{d x} [{(x^{\frac{3}{2}})}^{- 1}]$

Step 2.4.2

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 $g (x) = x^{\frac{3}{2}}$ .

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Step 2.4.2.1

To apply the Chain Rule, set $u$ as $x^{\frac{3}{2}}$ .

$12 x^{2} + 6 + \frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.4.2.2

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

$12 x^{2} + 6 - u^{- 2} \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.4.2.3

Replace all occurrences of $u$ with $x^{\frac{3}{2}}$ .

$12 x^{2} + 6 - {(x^{\frac{3}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.4.3

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

$12 x^{2} + 6 - {(x^{\frac{3}{2}})}^{- 2} (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 2.4.4

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

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Step 2.4.4.1

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

$12 x^{2} + 6 - x^{\frac{3}{2} \cdot - 2} (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 2.4.4.2

Cancel the common factor of $2$ .

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Step 2.4.4.2.1

Factor $2$ out of $- 2$ .

$12 x^{2} + 6 - x^{\frac{3}{2} \cdot (2 (- 1))} (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 2.4.4.2.2

Cancel the common factor.

$12 x^{2} + 6 - x^{\frac{3}{2} \cdot (2 \cdot - 1)} (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 2.4.4.2.3

Rewrite the expression.

$12 x^{2} + 6 - x^{3 \cdot - 1} (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 2.4.4.3

Multiply $3$ by $- 1$ .

$12 x^{2} + 6 - x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 2.4.5

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

$12 x^{2} + 6 - x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 2.4.6

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

$12 x^{2} + 6 - x^{- 3} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 2.4.7

Combine the numerators over the common denominator.

$12 x^{2} + 6 - x^{- 3} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})$

Step 2.4.8

Simplify the numerator.

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Step 2.4.8.1

Multiply $- 1$ by $2$ .

$12 x^{2} + 6 - x^{- 3} (\frac{3}{2} x^{\frac{3 - 2}{2}})$

Step 2.4.8.2

Subtract $2$ from $3$ .

$12 x^{2} + 6 - x^{- 3} (\frac{3}{2} x^{\frac{1}{2}})$

Step 2.4.9

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$12 x^{2} + 6 - x^{- 3} \frac{3 x^{\frac{1}{2}}}{2}$

Step 2.4.10

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

$12 x^{2} + 6 - \frac{3 x^{\frac{1}{2}} x^{- 3}}{2}$

Step 2.4.11

Multiply $x^{\frac{1}{2}}$ by $x^{- 3}$ by adding the exponents.

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Step 2.4.11.1

Move $x^{- 3}$ .

$12 x^{2} + 6 - \frac{3 (x^{- 3} x^{\frac{1}{2}})}{2}$

Step 2.4.11.2

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

$12 x^{2} + 6 - \frac{3 x^{- 3 + \frac{1}{2}}}{2}$

Step 2.4.11.3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$12 x^{2} + 6 - \frac{3 x^{- 3 \cdot \frac{2}{2} + \frac{1}{2}}}{2}$

Step 2.4.11.4

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

$12 x^{2} + 6 - \frac{3 x^{\frac{- 3 \cdot 2}{2} + \frac{1}{2}}}{2}$

Step 2.4.11.5

Combine the numerators over the common denominator.

$12 x^{2} + 6 - \frac{3 x^{\frac{- 3 \cdot 2 + 1}{2}}}{2}$

Step 2.4.11.6

Simplify the numerator.

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Step 2.4.11.6.1

Multiply $- 3$ by $2$ .

$12 x^{2} + 6 - \frac{3 x^{\frac{- 6 + 1}{2}}}{2}$

Step 2.4.11.6.2

Add $- 6$ and $1$ .

$12 x^{2} + 6 - \frac{3 x^{\frac{- 5}{2}}}{2}$

Step 2.4.11.7

Move the negative in front of the fraction.

$12 x^{2} + 6 - \frac{3 x^{- \frac{5}{2}}}{2}$

Step 2.4.12

Move $x^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 12 x^{2} + 6 - \frac{3}{2 x^{\frac{5}{2}}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $12 x^{2} + 6 - \frac{3}{2 x^{\frac{5}{2}}}$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [6] + \frac{d}{d x} [- \frac{3}{2 x^{\frac{5}{2}}}]$ .

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [6] + \frac{d}{d x} [- \frac{3}{2 x^{\frac{5}{2}}}]$

Step 3.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

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Step 3.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6] + \frac{d}{d x} [- \frac{3}{2 x^{\frac{5}{2}}}]$

Step 3.2.2

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

$12 (2 x) + \frac{d}{d x} [6] + \frac{d}{d x} [- \frac{3}{2 x^{\frac{5}{2}}}]$

Step 3.2.3

Multiply $2$ by $12$ .

$24 x + \frac{d}{d x} [6] + \frac{d}{d x} [- \frac{3}{2 x^{\frac{5}{2}}}]$

Step 3.3

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

$24 x + 0 + \frac{d}{d x} [- \frac{3}{2 x^{\frac{5}{2}}}]$

Step 3.4

Evaluate $\frac{d}{d x} [- \frac{3}{2 x^{\frac{5}{2}}}]$ .

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Step 3.4.1

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

$24 x + 0 - \frac{3}{2} \frac{d}{d x} [\frac{1}{x^{\frac{5}{2}}}]$

Step 3.4.2

Rewrite $\frac{1}{x^{\frac{5}{2}}}$ as ${(x^{\frac{5}{2}})}^{- 1}$ .

$24 x + 0 - \frac{3}{2} \frac{d}{d x} [{(x^{\frac{5}{2}})}^{- 1}]$

Step 3.4.3

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 $g (x) = x^{\frac{5}{2}}$ .

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Step 3.4.3.1

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

$24 x + 0 - \frac{3}{2} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.4.3.2

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

$24 x + 0 - \frac{3}{2} (- u^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.4.3.3

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

$24 x + 0 - \frac{3}{2} (- {(x^{\frac{5}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$24 x + 0 - \frac{3}{2} (- {(x^{\frac{5}{2}})}^{- 2} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.4.5

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

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Step 3.4.5.1

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

$24 x + 0 - \frac{3}{2} (- x^{\frac{5}{2} \cdot - 2} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.4.5.2

Cancel the common factor of $2$ .

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Step 3.4.5.2.1

Factor $2$ out of $- 2$ .

$24 x + 0 - \frac{3}{2} (- x^{\frac{5}{2} \cdot (2 (- 1))} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.4.5.2.2

Cancel the common factor.

$24 x + 0 - \frac{3}{2} (- x^{\frac{5}{2} \cdot (2 \cdot - 1)} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.4.5.2.3

Rewrite the expression.

$24 x + 0 - \frac{3}{2} (- x^{5 \cdot - 1} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.4.5.3

Multiply $5$ by $- 1$ .

$24 x + 0 - \frac{3}{2} (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.4.6

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

$24 x + 0 - \frac{3}{2} (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}))$

Step 3.4.7

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

$24 x + 0 - \frac{3}{2} (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 3.4.8

Combine the numerators over the common denominator.

$24 x + 0 - \frac{3}{2} (- x^{- 5} (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}))$

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Multiply $- 1$ by $2$ .

$24 x + 0 - \frac{3}{2} (- x^{- 5} (\frac{5}{2} x^{\frac{5 - 2}{2}}))$

Step 3.4.9.2

Subtract $2$ from $5$ .

$24 x + 0 - \frac{3}{2} (- x^{- 5} (\frac{5}{2} x^{\frac{3}{2}}))$

Step 3.4.10

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

$24 x + 0 - \frac{3}{2} (- x^{- 5} \frac{5 x^{\frac{3}{2}}}{2})$

Step 3.4.11

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

$24 x + 0 - \frac{3}{2} (- \frac{5 x^{\frac{3}{2}} x^{- 5}}{2})$

Step 3.4.12

Multiply $x^{\frac{3}{2}}$ by $x^{- 5}$ by adding the exponents.

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Step 3.4.12.1

Move $x^{- 5}$ .

$24 x + 0 - \frac{3}{2} (- \frac{5 (x^{- 5} x^{\frac{3}{2}})}{2})$

Step 3.4.12.2

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

$24 x + 0 - \frac{3}{2} (- \frac{5 x^{- 5 + \frac{3}{2}}}{2})$

Step 3.4.12.3

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

$24 x + 0 - \frac{3}{2} (- \frac{5 x^{- 5 \cdot \frac{2}{2} + \frac{3}{2}}}{2})$

Step 3.4.12.4

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

$24 x + 0 - \frac{3}{2} (- \frac{5 x^{\frac{- 5 \cdot 2}{2} + \frac{3}{2}}}{2})$

Step 3.4.12.5

Combine the numerators over the common denominator.

$24 x + 0 - \frac{3}{2} (- \frac{5 x^{\frac{- 5 \cdot 2 + 3}{2}}}{2})$

Step 3.4.12.6

Simplify the numerator.

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Step 3.4.12.6.1

Multiply $- 5$ by $2$ .

$24 x + 0 - \frac{3}{2} (- \frac{5 x^{\frac{- 10 + 3}{2}}}{2})$

Step 3.4.12.6.2

Add $- 10$ and $3$ .

$24 x + 0 - \frac{3}{2} (- \frac{5 x^{\frac{- 7}{2}}}{2})$

Step 3.4.12.7

Move the negative in front of the fraction.

$24 x + 0 - \frac{3}{2} (- \frac{5 x^{- \frac{7}{2}}}{2})$

Step 3.4.13

Move $x^{- \frac{7}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$24 x + 0 - \frac{3}{2} (- \frac{5}{2 x^{\frac{7}{2}}})$

Step 3.4.14

Multiply $- 1$ by $- 1$ .

$24 x + 0 + 1 (\frac{3}{2}) \frac{5}{2 x^{\frac{7}{2}}}$

Step 3.4.15

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

$24 x + 0 + \frac{3}{2} \cdot \frac{5}{2 x^{\frac{7}{2}}}$

Step 3.4.16

Multiply $\frac{3}{2}$ by $\frac{5}{2 x^{\frac{7}{2}}}$ .

$24 x + 0 + \frac{3 \cdot 5}{2 (2 x^{\frac{7}{2}})}$

Step 3.4.17

Multiply $3$ by $5$ .

$24 x + 0 + \frac{15}{2 (2 x^{\frac{7}{2}})}$

Step 3.4.18

Multiply $2$ by $2$ .

$24 x + 0 + \frac{15}{4 x^{\frac{7}{2}}}$

Step 3.5

Add $24 x$ and $0$ .

$f''' (x) = 24 x + \frac{15}{4 x^{\frac{7}{2}}}$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $24 x + \frac{15}{4 x^{\frac{7}{2}}}$ .

$24 x + \frac{15}{4 x^{\frac{7}{2}}}$