Calculus Examples

$y = {(1 + \frac{5}{x})}^{3}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 1 + \frac{5}{x}$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $1 + \frac{5}{x}$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [1 + \frac{5}{x}]$

Step 1.1.2

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

$3 u^{2} \frac{d}{d x} [1 + \frac{5}{x}]$

Step 1.1.3

Replace all occurrences of $u$ with $1 + \frac{5}{x}$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.2.3

Add $0$ and $\frac{d}{d x} [\frac{5}{x}]$ .

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

Step 1.2.4

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

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

Step 1.2.5

Simplify the expression.

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Step 1.2.5.1

Multiply $5$ by $3$ .

$15 {(1 + \frac{5}{x})}^{2} \frac{d}{d x} [\frac{1}{x}]$

Step 1.2.5.2

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

$15 {(1 + \frac{5}{x})}^{2} \frac{d}{d x} [x^{- 1}]$

Step 1.2.6

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

$15 {(1 + \frac{5}{x})}^{2} (- x^{- 2})$

Step 1.2.7

Multiply $- 1$ by $15$ .

$- 15 {(1 + \frac{5}{x})}^{2} x^{- 2}$

Step 1.3

Simplify.

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Step 1.3.1

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

$- 15 {(1 + \frac{5}{x})}^{2} \frac{1}{x^{2}}$

Step 1.3.2

Combine terms.

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Step 1.3.2.1

Combine $\frac{1}{x^{2}}$ and $- 15$ .

$\frac{- 15}{x^{2}} {(1 + \frac{5}{x})}^{2}$

Step 1.3.2.2

Combine $\frac{- 15}{x^{2}}$ and ${(1 + \frac{5}{x})}^{2}$ .

$\frac{- 15 {(1 + \frac{5}{x})}^{2}}{x^{2}}$

Step 1.3.2.3

Move the negative in front of the fraction.

$- \frac{15 {(1 + \frac{5}{x})}^{2}}{x^{2}}$

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Write $1$ as a fraction with a common denominator.

$- \frac{15 {(\frac{x}{x} + \frac{5}{x})}^{2}}{x^{2}}$

Step 1.3.3.2

Combine the numerators over the common denominator.

$- \frac{15 {(\frac{x + 5}{x})}^{2}}{x^{2}}$

Step 1.3.3.3

Apply the product rule to $\frac{x + 5}{x}$ .

$- \frac{15 \frac{{(x + 5)}^{2}}{x^{2}}}{x^{2}}$

Step 1.3.4

Combine $15$ and $\frac{{(x + 5)}^{2}}{x^{2}}$ .

$- \frac{\frac{15 {(x + 5)}^{2}}{x^{2}}}{x^{2}}$

Step 1.3.5

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{15 {(x + 5)}^{2}}{x^{2}} \cdot \frac{1}{x^{2}})$

Step 1.3.6

Combine.

$- \frac{15 {(x + 5)}^{2} \cdot 1}{x^{2} x^{2}}$

Step 1.3.7

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.3.7.1

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

$- \frac{15 {(x + 5)}^{2} \cdot 1}{x^{2 + 2}}$

Step 1.3.7.2

Add $2$ and $2$ .

$- \frac{15 {(x + 5)}^{2} \cdot 1}{x^{4}}$

Step 1.3.8

Multiply $15 {(x + 5)}^{2}$ by $1$ .

$f' (x) = - \frac{15 {(x + 5)}^{2}}{x^{4}}$

Step 2

Find the second derivative.

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Step 2.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- \frac{15 {(x + 5)}^{2}}{x^{4}}$ with respect to $x$ is $- 15 \frac{d}{d x} [\frac{{(x + 5)}^{2}}{x^{4}}]$ .

$- 15 \frac{d}{d x} [\frac{{(x + 5)}^{2}}{x^{4}}]$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = {(x + 5)}^{2}$ and $g (x) = x^{4}$ .

$- 15 \frac{x^{4} \frac{d}{d x} [{(x + 5)}^{2}] - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 2.3

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 2.3.1

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

$- 15 \frac{x^{4} \frac{d}{d x} [{(x + 5)}^{2}] - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{x^{4 \cdot 2}}$

Step 2.3.2

Multiply $4$ by $2$ .

$- 15 \frac{x^{4} \frac{d}{d x} [{(x + 5)}^{2}] - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x + 5$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x + 5$ .

$- 15 \frac{x^{4} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x + 5]) - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 2.4.2

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

$- 15 \frac{x^{4} (2 u \frac{d}{d x} [x + 5]) - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 2.4.3

Replace all occurrences of $u$ with $x + 5$ .

$- 15 \frac{x^{4} (2 (x + 5) \frac{d}{d x} [x + 5]) - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 2.5

Differentiate.

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Step 2.5.1

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

$- 15 \frac{x^{4} (2 (x + 5) (\frac{d}{d x} [x] + \frac{d}{d x} [5])) - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 2.5.2

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

$- 15 \frac{x^{4} (2 (x + 5) (1 + \frac{d}{d x} [5])) - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 2.5.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$- 15 \frac{x^{4} (2 (x + 5) (1 + 0)) - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $1$ and $0$ .

$- 15 \frac{x^{4} (2 (x + 5) \cdot 1) - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 2.5.4.2

Multiply $2$ by $1$ .

$- 15 \frac{x^{4} (2 (x + 5)) - {(x + 5)}^{2} \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 2.5.5

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

$- 15 \frac{x^{4} (2 (x + 5)) - {(x + 5)}^{2} (4 x^{3})}{x^{8}}$

Step 2.5.6

Combine fractions.

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Step 2.5.6.1

Multiply $4$ by $- 1$ .

$- 15 \frac{x^{4} (2 (x + 5)) - 4 {(x + 5)}^{2} x^{3}}{x^{8}}$

Step 2.5.6.2

Combine $- 15$ and $\frac{x^{4} (2 (x + 5)) - 4 {(x + 5)}^{2} x^{3}}{x^{8}}$ .

$\frac{- 15 (x^{4} (2 (x + 5)) - 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.5.6.3

Move the negative in front of the fraction.

$- \frac{15 (x^{4} (2 (x + 5)) - 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$- \frac{15 (x^{4} (2 x + 2 \cdot 5) - 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.2

Apply the distributive property.

$- \frac{15 (x^{4} (2 x) + x^{4} (2 \cdot 5) - 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.3

Apply the distributive property.

$- \frac{15 (x^{4} (2 x)) + 15 (x^{4} (2 \cdot 5)) + 15 (- 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.4

Simplify the numerator.

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Step 2.6.4.1

Simplify each term.

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Step 2.6.4.1.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 2.6.4.1.1.1

Move $x$ .

$- \frac{15 (x \cdot x^{4} \cdot 2) + 15 (x^{4} (2 \cdot 5)) + 15 (- 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.4.1.1.2

Multiply $x$ by $x^{4}$ .

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Step 2.6.4.1.1.2.1

Raise $x$ to the power of $1$ .

$- \frac{15 (x^{1} x^{4} \cdot 2) + 15 (x^{4} (2 \cdot 5)) + 15 (- 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.4.1.1.2.2

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

$- \frac{15 (x^{1 + 4} \cdot 2) + 15 (x^{4} (2 \cdot 5)) + 15 (- 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.4.1.1.3

Add $1$ and $4$ .

$- \frac{15 (x^{5} \cdot 2) + 15 (x^{4} (2 \cdot 5)) + 15 (- 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.4.1.2

Move $2$ to the left of $x^{5}$ .

$- \frac{15 (2 \cdot x^{5}) + 15 (x^{4} (2 \cdot 5)) + 15 (- 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.4.1.3

Multiply $2$ by $15$ .

$- \frac{30 x^{5} + 15 (x^{4} (2 \cdot 5)) + 15 (- 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.4.1.4

Multiply $2$ by $5$ .

$- \frac{30 x^{5} + 15 (x^{4} \cdot 10) + 15 (- 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.4.1.5

Move $10$ to the left of $x^{4}$ .

$- \frac{30 x^{5} + 15 (10 \cdot x^{4}) + 15 (- 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.4.1.6

Multiply $10$ by $15$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 {(x + 5)}^{2} x^{3})}{x^{8}}$

Step 2.6.4.1.7

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 ((x + 5) (x + 5)) x^{3})}{x^{8}}$

Step 2.6.4.1.8

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 2.6.4.1.8.1

Apply the distributive property.

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 (x (x + 5) + 5 (x + 5)) x^{3})}{x^{8}}$

Step 2.6.4.1.8.2

Apply the distributive property.

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 (x \cdot x + x \cdot 5 + 5 (x + 5)) x^{3})}{x^{8}}$

Step 2.6.4.1.8.3

Apply the distributive property.

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) x^{3})}{x^{8}}$

Step 2.6.4.1.9

Simplify and combine like terms.

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Step 2.6.4.1.9.1

Simplify each term.

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Step 2.6.4.1.9.1.1

Multiply $x$ by $x$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) x^{3})}{x^{8}}$

Step 2.6.4.1.9.1.2

Move $5$ to the left of $x$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) x^{3})}{x^{8}}$

Step 2.6.4.1.9.1.3

Multiply $5$ by $5$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 (x^{2} + 5 x + 5 x + 25) x^{3})}{x^{8}}$

Step 2.6.4.1.9.2

Add $5 x$ and $5 x$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 (x^{2} + 10 x + 25) x^{3})}{x^{8}}$

Step 2.6.4.1.10

Apply the distributive property.

$- \frac{30 x^{5} + 150 x^{4} + 15 ((- 4 x^{2} - 4 (10 x) - 4 \cdot 25) x^{3})}{x^{8}}$

Step 2.6.4.1.11

Simplify.

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Step 2.6.4.1.11.1

Multiply $10$ by $- 4$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 ((- 4 x^{2} - 40 x - 4 \cdot 25) x^{3})}{x^{8}}$

Step 2.6.4.1.11.2

Multiply $- 4$ by $25$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 ((- 4 x^{2} - 40 x - 100) x^{3})}{x^{8}}$

Step 2.6.4.1.12

Apply the distributive property.

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 x^{2} x^{3} - 40 x \cdot x^{3} - 100 x^{3})}{x^{8}}$

Step 2.6.4.1.13

Simplify.

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Step 2.6.4.1.13.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 2.6.4.1.13.1.1

Move $x^{3}$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 (x^{3} x^{2}) - 40 x \cdot x^{3} - 100 x^{3})}{x^{8}}$

Step 2.6.4.1.13.1.2

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

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 x^{3 + 2} - 40 x \cdot x^{3} - 100 x^{3})}{x^{8}}$

Step 2.6.4.1.13.1.3

Add $3$ and $2$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 x^{5} - 40 x \cdot x^{3} - 100 x^{3})}{x^{8}}$

Step 2.6.4.1.13.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.6.4.1.13.2.1

Move $x^{3}$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 x^{5} - 40 (x^{3} x) - 100 x^{3})}{x^{8}}$

Step 2.6.4.1.13.2.2

Multiply $x^{3}$ by $x$ .

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Step 2.6.4.1.13.2.2.1

Raise $x$ to the power of $1$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 x^{5} - 40 (x^{3} x^{1}) - 100 x^{3})}{x^{8}}$

Step 2.6.4.1.13.2.2.2

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

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 x^{5} - 40 x^{3 + 1} - 100 x^{3})}{x^{8}}$

Step 2.6.4.1.13.2.3

Add $3$ and $1$ .

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 x^{5} - 40 x^{4} - 100 x^{3})}{x^{8}}$

Step 2.6.4.1.14

Apply the distributive property.

$- \frac{30 x^{5} + 150 x^{4} + 15 (- 4 x^{5}) + 15 (- 40 x^{4}) + 15 (- 100 x^{3})}{x^{8}}$

Step 2.6.4.1.15

Simplify.

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Step 2.6.4.1.15.1

Multiply $- 4$ by $15$ .

$- \frac{30 x^{5} + 150 x^{4} - 60 x^{5} + 15 (- 40 x^{4}) + 15 (- 100 x^{3})}{x^{8}}$

Step 2.6.4.1.15.2

Multiply $- 40$ by $15$ .

$- \frac{30 x^{5} + 150 x^{4} - 60 x^{5} - 600 x^{4} + 15 (- 100 x^{3})}{x^{8}}$

Step 2.6.4.1.15.3

Multiply $- 100$ by $15$ .

$- \frac{30 x^{5} + 150 x^{4} - 60 x^{5} - 600 x^{4} - 1500 x^{3}}{x^{8}}$

Step 2.6.4.2

Subtract $60 x^{5}$ from $30 x^{5}$ .

$- \frac{- 30 x^{5} + 150 x^{4} - 600 x^{4} - 1500 x^{3}}{x^{8}}$

Step 2.6.4.3

Subtract $600 x^{4}$ from $150 x^{4}$ .

$- \frac{- 30 x^{5} - 450 x^{4} - 1500 x^{3}}{x^{8}}$

Step 2.6.5

Simplify the numerator.

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Step 2.6.5.1

Factor $30 x^{3}$ out of $- 30 x^{5} - 450 x^{4} - 1500 x^{3}$ .

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Step 2.6.5.1.1

Factor $30 x^{3}$ out of $- 30 x^{5}$ .

$- \frac{30 x^{3} (- x^{2}) - 450 x^{4} - 1500 x^{3}}{x^{8}}$

Step 2.6.5.1.2

Factor $30 x^{3}$ out of $- 450 x^{4}$ .

$- \frac{30 x^{3} (- x^{2}) + 30 x^{3} (- 15 x) - 1500 x^{3}}{x^{8}}$

Step 2.6.5.1.3

Factor $30 x^{3}$ out of $- 1500 x^{3}$ .

$- \frac{30 x^{3} (- x^{2}) + 30 x^{3} (- 15 x) + 30 x^{3} (- 50)}{x^{8}}$

Step 2.6.5.1.4

Factor $30 x^{3}$ out of $30 x^{3} (- x^{2}) + 30 x^{3} (- 15 x)$ .

$- \frac{30 x^{3} (- x^{2} - 15 x) + 30 x^{3} (- 50)}{x^{8}}$

Step 2.6.5.1.5

Factor $30 x^{3}$ out of $30 x^{3} (- x^{2} - 15 x) + 30 x^{3} (- 50)$ .

$- \frac{30 x^{3} (- x^{2} - 15 x - 50)}{x^{8}}$

Step 2.6.5.2

Factor by grouping.

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Step 2.6.5.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 50 = 50$ and whose sum is $b = - 15$ .

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Step 2.6.5.2.1.1

Factor $- 15$ out of $- 15 x$ .

$- \frac{30 x^{3} (- x^{2} - 15 (x) - 50)}{x^{8}}$

Step 2.6.5.2.1.2

Rewrite $- 15$ as $- 5$ plus $- 10$

$- \frac{30 x^{3} (- x^{2} + (- 5 - 10) x - 50)}{x^{8}}$

Step 2.6.5.2.1.3

Apply the distributive property.

$- \frac{30 x^{3} (- x^{2} - 5 x - 10 x - 50)}{x^{8}}$

Step 2.6.5.2.2

Factor out the greatest common factor from each group.

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Step 2.6.5.2.2.1

Group the first two terms and the last two terms.

$- \frac{30 x^{3} ((- x^{2} - 5 x) - 10 x - 50)}{x^{8}}$

Step 2.6.5.2.2.2

Factor out the greatest common factor (GCF) from each group.

$- \frac{30 x^{3} (x (- x - 5) + 10 (- x - 5))}{x^{8}}$

Step 2.6.5.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 5$ .

$- \frac{30 x^{3} ((- x - 5) (x + 10))}{x^{8}}$

$- \frac{30 x^{3} (- x - 5) (x + 10)}{x^{8}}$

Step 2.6.6

Cancel the common factor of $x^{3}$ and $x^{8}$ .

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Step 2.6.6.1

Factor $x^{3}$ out of $30 x^{3} (- x - 5) (x + 10)$ .

$- \frac{x^{3} ((30 (- x - 5)) (x + 10))}{x^{8}}$

Step 2.6.6.2

Cancel the common factors.

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Step 2.6.6.2.1

Factor $x^{3}$ out of $x^{8}$ .

$- \frac{x^{3} ((30 (- x - 5)) (x + 10))}{x^{3} x^{5}}$

Step 2.6.6.2.2

Cancel the common factor.

$- \frac{x^{3} ((30 (- x - 5)) (x + 10))}{x^{3} x^{5}}$

Step 2.6.6.2.3

Rewrite the expression.

$- \frac{(30 (- x - 5)) (x + 10)}{x^{5}}$

Step 2.6.7

Factor $- 1$ out of $- x$ .

$- \frac{30 (- (x) - 5) (x + 10)}{x^{5}}$

Step 2.6.8

Rewrite $- 5$ as $- 1 (5)$ .

$- \frac{30 (- (x) - 1 (5)) (x + 10)}{x^{5}}$

Step 2.6.9

Factor $- 1$ out of $- (x) - 1 (5)$ .

$- \frac{30 (- (x + 5)) (x + 10)}{x^{5}}$

Step 2.6.10

Rewrite $- (x + 5)$ as $- 1 (x + 5)$ .

$- \frac{30 (- 1 (x + 5)) (x + 10)}{x^{5}}$

Step 2.6.11

Move the negative in front of the fraction.

$- - \frac{(30 (x + 5)) (x + 10)}{x^{5}}$

Step 2.6.12

Multiply $- 1$ by $- 1$ .

$1 \frac{(30 (x + 5)) (x + 10)}{x^{5}}$

Step 2.6.13

Multiply $\frac{(30 (x + 5)) (x + 10)}{x^{5}}$ by $1$ .

$\frac{(30 (x + 5)) (x + 10)}{x^{5}}$

Step 2.6.14

Reorder factors in $\frac{(30 (x + 5)) (x + 10)}{x^{5}}$ .

$f'' (x) = \frac{30 (x + 5) (x + 10)}{x^{5}}$

Step 3

Find the third derivative.

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Step 3.1

Since $30$ is constant with respect to $x$ , the derivative of $\frac{30 (x + 5) (x + 10)}{x^{5}}$ with respect to $x$ is $30 \frac{d}{d x} [\frac{(x + 5) (x + 10)}{x^{5}}]$ .

$30 \frac{d}{d x} [\frac{(x + 5) (x + 10)}{x^{5}}]$

Step 3.2

$30 \frac{x^{5} \frac{d}{d x} [(x + 5) (x + 10)] - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{{(x^{5})}^{2}}$

Step 3.3

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

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Step 3.3.1

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

$30 \frac{x^{5} \frac{d}{d x} [(x + 5) (x + 10)] - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{5 \cdot 2}}$

Step 3.3.2

Multiply $5$ by $2$ .

$30 \frac{x^{5} \frac{d}{d x} [(x + 5) (x + 10)] - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 5$ and $g (x) = x + 10$ .

$30 \frac{x^{5} ((x + 5) \frac{d}{d x} [x + 10] + (x + 10) \frac{d}{d x} [x + 5]) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5

Differentiate.

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Step 3.5.1

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

$30 \frac{x^{5} ((x + 5) (\frac{d}{d x} [x] + \frac{d}{d x} [10]) + (x + 10) \frac{d}{d x} [x + 5]) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.2

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

$30 \frac{x^{5} ((x + 5) (1 + \frac{d}{d x} [10]) + (x + 10) \frac{d}{d x} [x + 5]) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.3

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$30 \frac{x^{5} ((x + 5) (1 + 0) + (x + 10) \frac{d}{d x} [x + 5]) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.4

Simplify the expression.

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Step 3.5.4.1

Add $1$ and $0$ .

$30 \frac{x^{5} ((x + 5) \cdot 1 + (x + 10) \frac{d}{d x} [x + 5]) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.4.2

Multiply $x + 5$ by $1$ .

$30 \frac{x^{5} (x + 5 + (x + 10) \frac{d}{d x} [x + 5]) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.5

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

$30 \frac{x^{5} (x + 5 + (x + 10) (\frac{d}{d x} [x] + \frac{d}{d x} [5])) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.6

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

$30 \frac{x^{5} (x + 5 + (x + 10) (1 + \frac{d}{d x} [5])) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.7

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$30 \frac{x^{5} (x + 5 + (x + 10) (1 + 0)) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.8

Simplify by adding terms.

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Step 3.5.8.1

Add $1$ and $0$ .

$30 \frac{x^{5} (x + 5 + (x + 10) \cdot 1) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.8.2

Multiply $x + 10$ by $1$ .

$30 \frac{x^{5} (x + 5 + x + 10) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.8.3

Add $x$ and $x$ .

$30 \frac{x^{5} (2 x + 5 + 10) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.8.4

Add $5$ and $10$ .

$30 \frac{x^{5} (2 x + 15) - (x + 5) (x + 10) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.5.9

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

$30 \frac{x^{5} (2 x + 15) - (x + 5) (x + 10) (5 x^{4})}{x^{10}}$

Step 3.5.10

Simplify with factoring out.

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Step 3.5.10.1

Multiply $5$ by $- 1$ .

$30 \frac{x^{5} (2 x + 15) - 5 (x + 5) (x + 10) x^{4}}{x^{10}}$

Step 3.5.10.2

Factor $x^{4}$ out of $x^{5} (2 x + 15) - 5 (x + 5) (x + 10) x^{4}$ .

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Step 3.5.10.2.1

Factor $x^{4}$ out of $x^{5} (2 x + 15)$ .

$30 \frac{x^{4} (x (2 x + 15)) - 5 (x + 5) (x + 10) x^{4}}{x^{10}}$

Step 3.5.10.2.2

Factor $x^{4}$ out of $- 5 (x + 5) (x + 10) x^{4}$ .

$30 \frac{x^{4} (x (2 x + 15)) + x^{4} (- 5 (x + 5) (x + 10))}{x^{10}}$

Step 3.5.10.2.3

Factor $x^{4}$ out of $x^{4} (x (2 x + 15)) + x^{4} (- 5 (x + 5) (x + 10))$ .

$30 \frac{x^{4} (x (2 x + 15) - 5 (x + 5) (x + 10))}{x^{10}}$

Step 3.6

Cancel the common factors.

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Step 3.6.1

Factor $x^{4}$ out of $x^{10}$ .

$30 \frac{x^{4} (x (2 x + 15) - 5 (x + 5) (x + 10))}{x^{4} x^{6}}$

Step 3.6.2

Cancel the common factor.

$30 \frac{x^{4} (x (2 x + 15) - 5 (x + 5) (x + 10))}{x^{4} x^{6}}$

Step 3.6.3

Rewrite the expression.

$30 \frac{x (2 x + 15) - 5 (x + 5) (x + 10)}{x^{6}}$

Step 3.7

Combine $30$ and $\frac{x (2 x + 15) - 5 (x + 5) (x + 10)}{x^{6}}$ .

$\frac{30 (x (2 x + 15) - 5 (x + 5) (x + 10))}{x^{6}}$

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

$\frac{30 (x (2 x) + x \cdot 15 - 5 (x + 5) (x + 10))}{x^{6}}$

Step 3.8.2

Apply the distributive property.

$\frac{30 (x (2 x) + x \cdot 15 + (- 5 x - 5 \cdot 5) (x + 10))}{x^{6}}$

Step 3.8.3

Apply the distributive property.

$\frac{30 (x (2 x)) + 30 (x \cdot 15) + 30 ((- 5 x - 5 \cdot 5) (x + 10))}{x^{6}}$

Step 3.8.4

Simplify the numerator.

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Step 3.8.4.1

Simplify each term.

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Step 3.8.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.8.4.1.1.1

Move $x$ .

$\frac{30 (x \cdot x \cdot 2) + 30 (x \cdot 15) + 30 ((- 5 x - 5 \cdot 5) (x + 10))}{x^{6}}$

Step 3.8.4.1.1.2

Multiply $x$ by $x$ .

$\frac{30 (x^{2} \cdot 2) + 30 (x \cdot 15) + 30 ((- 5 x - 5 \cdot 5) (x + 10))}{x^{6}}$

Step 3.8.4.1.2

Move $2$ to the left of $x^{2}$ .

$\frac{30 (2 \cdot x^{2}) + 30 (x \cdot 15) + 30 ((- 5 x - 5 \cdot 5) (x + 10))}{x^{6}}$

Step 3.8.4.1.3

Multiply $2$ by $30$ .

$\frac{60 x^{2} + 30 (x \cdot 15) + 30 ((- 5 x - 5 \cdot 5) (x + 10))}{x^{6}}$

Step 3.8.4.1.4

Move $15$ to the left of $x$ .

$\frac{60 x^{2} + 30 (15 \cdot x) + 30 ((- 5 x - 5 \cdot 5) (x + 10))}{x^{6}}$

Step 3.8.4.1.5

Multiply $15$ by $30$ .

$\frac{60 x^{2} + 450 x + 30 ((- 5 x - 5 \cdot 5) (x + 10))}{x^{6}}$

Step 3.8.4.1.6

Multiply $- 5$ by $5$ .

$\frac{60 x^{2} + 450 x + 30 ((- 5 x - 25) (x + 10))}{x^{6}}$

Step 3.8.4.1.7

Expand $(- 5 x - 25) (x + 10)$ using the FOIL Method.

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Step 3.8.4.1.7.1

Apply the distributive property.

$\frac{60 x^{2} + 450 x + 30 (- 5 x (x + 10) - 25 (x + 10))}{x^{6}}$

Step 3.8.4.1.7.2

Apply the distributive property.

$\frac{60 x^{2} + 450 x + 30 (- 5 x \cdot x - 5 x \cdot 10 - 25 (x + 10))}{x^{6}}$

Step 3.8.4.1.7.3

Apply the distributive property.

$\frac{60 x^{2} + 450 x + 30 (- 5 x \cdot x - 5 x \cdot 10 - 25 x - 25 \cdot 10)}{x^{6}}$

Step 3.8.4.1.8

Simplify and combine like terms.

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Step 3.8.4.1.8.1

Simplify each term.

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Step 3.8.4.1.8.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.8.4.1.8.1.1.1

Move $x$ .

$\frac{60 x^{2} + 450 x + 30 (- 5 (x \cdot x) - 5 x \cdot 10 - 25 x - 25 \cdot 10)}{x^{6}}$

Step 3.8.4.1.8.1.1.2

Multiply $x$ by $x$ .

$\frac{60 x^{2} + 450 x + 30 (- 5 x^{2} - 5 x \cdot 10 - 25 x - 25 \cdot 10)}{x^{6}}$

Step 3.8.4.1.8.1.2

Multiply $10$ by $- 5$ .

$\frac{60 x^{2} + 450 x + 30 (- 5 x^{2} - 50 x - 25 x - 25 \cdot 10)}{x^{6}}$

Step 3.8.4.1.8.1.3

Multiply $- 25$ by $10$ .

$\frac{60 x^{2} + 450 x + 30 (- 5 x^{2} - 50 x - 25 x - 250)}{x^{6}}$

Step 3.8.4.1.8.2

Subtract $25 x$ from $- 50 x$ .

$\frac{60 x^{2} + 450 x + 30 (- 5 x^{2} - 75 x - 250)}{x^{6}}$

Step 3.8.4.1.9

Apply the distributive property.

$\frac{60 x^{2} + 450 x + 30 (- 5 x^{2}) + 30 (- 75 x) + 30 \cdot - 250}{x^{6}}$

Step 3.8.4.1.10

Simplify.

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Step 3.8.4.1.10.1

Multiply $- 5$ by $30$ .

$\frac{60 x^{2} + 450 x - 150 x^{2} + 30 (- 75 x) + 30 \cdot - 250}{x^{6}}$

Step 3.8.4.1.10.2

Multiply $- 75$ by $30$ .

$\frac{60 x^{2} + 450 x - 150 x^{2} - 2250 x + 30 \cdot - 250}{x^{6}}$

Step 3.8.4.1.10.3

Multiply $30$ by $- 250$ .

$\frac{60 x^{2} + 450 x - 150 x^{2} - 2250 x - 7500}{x^{6}}$

Step 3.8.4.2

Subtract $150 x^{2}$ from $60 x^{2}$ .

$\frac{- 90 x^{2} + 450 x - 2250 x - 7500}{x^{6}}$

Step 3.8.4.3

Subtract $2250 x$ from $450 x$ .

$\frac{- 90 x^{2} - 1800 x - 7500}{x^{6}}$

Step 3.8.5

Factor $30$ out of $- 90 x^{2} - 1800 x - 7500$ .

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Step 3.8.5.1

Factor $30$ out of $- 90 x^{2}$ .

$\frac{30 (- 3 x^{2}) - 1800 x - 7500}{x^{6}}$

Step 3.8.5.2

Factor $30$ out of $- 1800 x$ .

$\frac{30 (- 3 x^{2}) + 30 (- 60 x) - 7500}{x^{6}}$

Step 3.8.5.3

Factor $30$ out of $- 7500$ .

$\frac{30 (- 3 x^{2}) + 30 (- 60 x) + 30 (- 250)}{x^{6}}$

Step 3.8.5.4

Factor $30$ out of $30 (- 3 x^{2}) + 30 (- 60 x)$ .

$\frac{30 (- 3 x^{2} - 60 x) + 30 (- 250)}{x^{6}}$

Step 3.8.5.5

Factor $30$ out of $30 (- 3 x^{2} - 60 x) + 30 (- 250)$ .

$\frac{30 (- 3 x^{2} - 60 x - 250)}{x^{6}}$

Step 3.8.6

Factor $- 1$ out of $- 3 x^{2}$ .

$\frac{30 (- (3 x^{2}) - 60 x - 250)}{x^{6}}$

Step 3.8.7

Factor $- 1$ out of $- 60 x$ .

$\frac{30 (- (3 x^{2}) - (60 x) - 250)}{x^{6}}$

Step 3.8.8

Factor $- 1$ out of $- (3 x^{2}) - (60 x)$ .

$\frac{30 (- (3 x^{2} + 60 x) - 250)}{x^{6}}$

Step 3.8.9

Rewrite $- 250$ as $- 1 (250)$ .

$\frac{30 (- (3 x^{2} + 60 x) - 1 (250))}{x^{6}}$

Step 3.8.10

Factor $- 1$ out of $- (3 x^{2} + 60 x) - 1 (250)$ .

$\frac{30 (- (3 x^{2} + 60 x + 250))}{x^{6}}$

Step 3.8.11

Rewrite $- (3 x^{2} + 60 x + 250)$ as $- 1 (3 x^{2} + 60 x + 250)$ .

$\frac{30 (- 1 (3 x^{2} + 60 x + 250))}{x^{6}}$

Step 3.8.12

Move the negative in front of the fraction.

$f''' (x) = - \frac{30 (3 x^{2} + 60 x + 250)}{x^{6}}$