Calculus Examples

$y = \frac{{(x - 5)}^{7}}{{(x - 4)}^{6}}$

Step 1

Write $y = \frac{{(x - 5)}^{7}}{{(x - 4)}^{6}}$ as a function.

$f (x) = \frac{{(x - 5)}^{7}}{{(x - 4)}^{6}}$

Step 2

Find the first derivative of the function.

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Step 2.1

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 $f (x) = {(x - 5)}^{7}$ and $g (x) = {(x - 4)}^{6}$ .

$\frac{{(x - 4)}^{6} \frac{d}{d x} [{(x - 5)}^{7}] - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{({(x - 4)}^{6})}^{2}}$

Step 2.2

Multiply the exponents in ${({(x - 4)}^{6})}^{2}$ .

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Step 2.2.1

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

$\frac{{(x - 4)}^{6} \frac{d}{d x} [{(x - 5)}^{7}] - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{6 \cdot 2}}$

Step 2.2.2

Multiply $6$ by $2$ .

$\frac{{(x - 4)}^{6} \frac{d}{d x} [{(x - 5)}^{7}] - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = x - 5$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $x - 5$ .

$\frac{{(x - 4)}^{6} (\frac{d}{d u_{1}} [{u_{1}}^{7}] \frac{d}{d x} [x - 5]) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 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 $n = 7$ .

$\frac{{(x - 4)}^{6} (7 {u_{1}}^{6} \frac{d}{d x} [x - 5]) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $x - 5$ .

$\frac{{(x - 4)}^{6} (7 {(x - 5)}^{6} \frac{d}{d x} [x - 5]) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 2.4

Differentiate.

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Step 2.4.1

Move $7$ to the left of ${(x - 4)}^{6}$ .

$\frac{7 \cdot {(x - 4)}^{6} {(x - 5)}^{6} \frac{d}{d x} [x - 5] - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 2.4.2

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 2.4.3

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} (1 + \frac{d}{d x} [- 5]) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 2.4.4

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} (1 + 0) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 2.4.5

Simplify the expression.

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Step 2.4.5.1

Add $1$ and $0$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} \cdot 1 - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 2.4.5.2

Multiply $7$ by $1$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = x - 4$ .

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Step 2.5.1

To apply the Chain Rule, set $u_{2}$ as $x - 4$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - {(x - 5)}^{7} (\frac{d}{d u_{2}} [{u_{2}}^{6}] \frac{d}{d x} [x - 4])}{{(x - 4)}^{12}}$

Step 2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 6$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - {(x - 5)}^{7} (6 {u_{2}}^{5} \frac{d}{d x} [x - 4])}{{(x - 4)}^{12}}$

Step 2.5.3

Replace all occurrences of $u_{2}$ with $x - 4$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - {(x - 5)}^{7} (6 {(x - 4)}^{5} \frac{d}{d x} [x - 4])}{{(x - 4)}^{12}}$

Step 2.6

Differentiate.

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Step 2.6.1

Multiply $6$ by $- 1$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} ({(x - 4)}^{5} \frac{d}{d x} [x - 4])}{{(x - 4)}^{12}}$

Step 2.6.2

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} ({(x - 4)}^{5} (\frac{d}{d x} [x] + \frac{d}{d x} [- 4]))}{{(x - 4)}^{12}}$

Step 2.6.3

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} ({(x - 4)}^{5} (1 + \frac{d}{d x} [- 4]))}{{(x - 4)}^{12}}$

Step 2.6.4

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} ({(x - 4)}^{5} (1 + 0))}{{(x - 4)}^{12}}$

Step 2.6.5

Simplify the expression.

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Step 2.6.5.1

Add $1$ and $0$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} ({(x - 4)}^{5} \cdot 1)}{{(x - 4)}^{12}}$

Step 2.6.5.2

Multiply ${(x - 4)}^{5}$ by $1$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} {(x - 4)}^{5}}{{(x - 4)}^{12}}$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Factor ${(x - 4)}^{5} {(x - 5)}^{6}$ out of $7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} {(x - 4)}^{5}$ .

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Step 2.7.1.1.1

Factor ${(x - 4)}^{5} {(x - 5)}^{6}$ out of $7 {(x - 4)}^{6} {(x - 5)}^{6}$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 (x - 4)) - 6 {(x - 5)}^{7} {(x - 4)}^{5}}{{(x - 4)}^{12}}$

Step 2.7.1.1.2

Factor ${(x - 4)}^{5} {(x - 5)}^{6}$ out of $- 6 {(x - 5)}^{7} {(x - 4)}^{5}$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 (x - 4)) + {(x - 4)}^{5} {(x - 5)}^{6} (- 6 (x - 5))}{{(x - 4)}^{12}}$

Step 2.7.1.1.3

Factor ${(x - 4)}^{5} {(x - 5)}^{6}$ out of ${(x - 4)}^{5} {(x - 5)}^{6} (7 (x - 4)) + {(x - 4)}^{5} {(x - 5)}^{6} (- 6 (x - 5))$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 (x - 4) - 6 (x - 5))}{{(x - 4)}^{12}}$

Step 2.7.1.2

Apply the distributive property.

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 x + 7 \cdot - 4 - 6 (x - 5))}{{(x - 4)}^{12}}$

Step 2.7.1.3

Multiply $7$ by $- 4$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 x - 28 - 6 (x - 5))}{{(x - 4)}^{12}}$

Step 2.7.1.4

Apply the distributive property.

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 x - 28 - 6 x - 6 \cdot - 5)}{{(x - 4)}^{12}}$

Step 2.7.1.5

Multiply $- 6$ by $- 5$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 x - 28 - 6 x + 30)}{{(x - 4)}^{12}}$

Step 2.7.1.6

Subtract $6 x$ from $7 x$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (x - 28 + 30)}{{(x - 4)}^{12}}$

Step 2.7.1.7

Add $- 28$ and $30$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{12}}$

Step 2.7.2

Cancel the common factor of ${(x - 4)}^{5}$ and ${(x - 4)}^{12}$ .

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Step 2.7.2.1

Factor ${(x - 4)}^{5}$ out of ${(x - 4)}^{5} {(x - 5)}^{6} (x + 2)$ .

$\frac{{(x - 4)}^{5} ({(x - 5)}^{6} (x + 2))}{{(x - 4)}^{12}}$

Step 2.7.2.2

Cancel the common factors.

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Step 2.7.2.2.1

Factor ${(x - 4)}^{5}$ out of ${(x - 4)}^{12}$ .

$\frac{{(x - 4)}^{5} ({(x - 5)}^{6} (x + 2))}{{(x - 4)}^{5} {(x - 4)}^{7}}$

Step 2.7.2.2.2

Cancel the common factor.

$\frac{{(x - 4)}^{5} ({(x - 5)}^{6} (x + 2))}{{(x - 4)}^{5} {(x - 4)}^{7}}$

Step 2.7.2.2.3

Rewrite the expression.

$\frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}}$

Step 3

Find the second derivative of the function.

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Step 3.1

$f'' (x) = \frac{{(x - 4)}^{7} \frac{d}{d x} ({(x - 5)}^{6} (x + 2)) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{({(x - 4)}^{7})}^{2}}$

Step 3.2

Multiply the exponents in ${({(x - 4)}^{7})}^{2}$ .

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Step 3.2.1

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

$f'' (x) = \frac{{(x - 4)}^{7} \frac{d}{d x} ({(x - 5)}^{6} (x + 2)) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{7 \cdot 2}}$

Step 3.2.2

Multiply $7$ by $2$ .

$f'' (x) = \frac{{(x - 4)}^{7} \frac{d}{d x} ({(x - 5)}^{6} (x + 2)) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.3

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 $f (x) = {(x - 5)}^{6}$ and $g (x) = x + 2$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} \frac{d}{d x} (x + 2) + (x + 2) \frac{d}{d x} ({(x - 5)}^{6})) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.4

Differentiate.

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Step 3.4.1

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

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} (\frac{d}{d x} (x) + \frac{d}{d x} (2)) + (x + 2) \frac{d}{d x} ({(x - 5)}^{6})) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.4.2

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

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} (1 + \frac{d}{d x} (2)) + (x + 2) \frac{d}{d x} ({(x - 5)}^{6})) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.4.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} (1 + 0) + (x + 2) \frac{d}{d x} ({(x - 5)}^{6})) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.4.4

Simplify the expression.

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Step 3.4.4.1

Add $1$ and $0$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} \cdot 1 + (x + 2) \frac{d}{d x} ({(x - 5)}^{6})) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.4.4.2

Multiply ${(x - 5)}^{6}$ by $1$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + (x + 2) \frac{d}{d x} ({(x - 5)}^{6})) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = x - 5$ .

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Step 3.5.1

To apply the Chain Rule, set $u_{1}$ as $x - 5$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + (x + 2) (\frac{d}{d u (1)} ({u_{1}}^{6}) \frac{d}{d x} (x - 5))) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 6$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + (x + 2) (6 {u_{1}}^{5} \frac{d}{d x} (x - 5))) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.5.3

Replace all occurrences of $u_{1}$ with $x - 5$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + (x + 2) (6 {(x - 5)}^{5} \frac{d}{d x} (x - 5))) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.6

Differentiate.

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Step 3.6.1

Move $6$ to the left of $x + 2$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + 6 \cdot ((x + 2) {(x - 5)}^{5}) \frac{d}{d x} (x - 5)) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.6.2

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

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5} (\frac{d}{d x} (x) + \frac{d}{d x} (- 5))) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.6.3

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

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5} (1 + \frac{d}{d x} (- 5))) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.6.4

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

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5} (1 + 0)) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.6.5

Simplify the expression.

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Step 3.6.5.1

Add $1$ and $0$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5} \cdot 1) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.6.5.2

Multiply $6$ by $1$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - {(x - 5)}^{6} (x + 2) \frac{d}{d x} {(x - 4)}^{7}}{{(x - 4)}^{14}}$

Step 3.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = x - 4$ .

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Step 3.7.1

To apply the Chain Rule, set $u_{2}$ as $x - 4$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - {(x - 5)}^{6} (x + 2) (\frac{d}{d u (2)} ({u_{2}}^{7}) \frac{d}{d x} (x - 4))}{{(x - 4)}^{14}}$

Step 3.7.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 7$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - {(x - 5)}^{6} (x + 2) (7 {u_{2}}^{6} \frac{d}{d x} (x - 4))}{{(x - 4)}^{14}}$

Step 3.7.3

Replace all occurrences of $u_{2}$ with $x - 4$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - {(x - 5)}^{6} (x + 2) (7 {(x - 4)}^{6} \frac{d}{d x} (x - 4))}{{(x - 4)}^{14}}$

Step 3.8

Simplify with factoring out.

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Step 3.8.1

Multiply $7$ by $- 1$ .

$f'' (x) = \frac{{(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2) ({(x - 4)}^{6} \frac{d}{d x} (x - 4))}{{(x - 4)}^{14}}$

Step 3.8.2

Factor ${(x - 4)}^{6}$ out of ${(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2) ({(x - 4)}^{6} \frac{d}{d x} [x - 4])$ .

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Step 3.8.2.1

Factor ${(x - 4)}^{6}$ out of ${(x - 4)}^{7} ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5})$ .

$f'' (x) = \frac{{(x - 4)}^{6} ((x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5})) - 7 {(x - 5)}^{6} (x + 2) ({(x - 4)}^{6} \frac{d}{d x} (x - 4))}{{(x - 4)}^{14}}$

Step 3.8.2.2

Factor ${(x - 4)}^{6}$ out of $- 7 {(x - 5)}^{6} (x + 2) ({(x - 4)}^{6} \frac{d}{d x} [x - 4])$ .

$f'' (x) = \frac{{(x - 4)}^{6} ((x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5})) + {(x - 4)}^{6} (- 7 {(x - 5)}^{6} (x + 2) (\frac{d}{d x} (x - 4)))}{{(x - 4)}^{14}}$

Step 3.8.2.3

Factor ${(x - 4)}^{6}$ out of ${(x - 4)}^{6} ((x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5})) + {(x - 4)}^{6} (- 7 {(x - 5)}^{6} (x + 2) (\frac{d}{d x} [x - 4]))$ .

$f'' (x) = \frac{{(x - 4)}^{6} ((x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2) (\frac{d}{d x} (x - 4)))}{{(x - 4)}^{14}}$

Step 3.9

Cancel the common factors.

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Step 3.9.1

Factor ${(x - 4)}^{6}$ out of ${(x - 4)}^{14}$ .

$f'' (x) = \frac{{(x - 4)}^{6} ((x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2) (\frac{d}{d x} (x - 4)))}{{(x - 4)}^{6} {(x - 4)}^{8}}$

Step 3.9.2

Cancel the common factor.

$f'' (x) = \frac{{(x - 4)}^{6} ((x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2) (\frac{d}{d x} (x - 4)))}{{(x - 4)}^{6} {(x - 4)}^{8}}$

Step 3.9.3

Rewrite the expression.

$f'' (x) = \frac{(x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2) (\frac{d}{d x} (x - 4))}{{(x - 4)}^{8}}$

Step 3.10

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

$f'' (x) = \frac{(x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2) (\frac{d}{d x} (x) + \frac{d}{d x} (- 4))}{{(x - 4)}^{8}}$

Step 3.11

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

$f'' (x) = \frac{(x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2) (1 + \frac{d}{d x} (- 4))}{{(x - 4)}^{8}}$

Step 3.12

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

$f'' (x) = \frac{(x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2) (1 + 0)}{{(x - 4)}^{8}}$

Step 3.13

Simplify the expression.

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Step 3.13.1

Add $1$ and $0$ .

$f'' (x) = \frac{(x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2) \cdot 1}{{(x - 4)}^{8}}$

Step 3.13.2

Multiply $- 7$ by $1$ .

$f'' (x) = \frac{(x - 4) ({(x - 5)}^{6} + 6 (x + 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14

Simplify.

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Step 3.14.1

Apply the distributive property.

$f'' (x) = \frac{(x - 4) ({(x - 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2

Simplify the numerator.

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Step 3.14.2.1

Simplify each term.

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Step 3.14.2.1.1

Use the Binomial Theorem.

$f'' (x) = \frac{(x - 4) (x^{6} + 6 x^{5} \cdot - 5 + 15 x^{4} {(- 5)}^{2} + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.2

Simplify each term.

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Step 3.14.2.1.2.1

Multiply $- 5$ by $6$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 15 x^{4} {(- 5)}^{2} + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.2.2

Raise $- 5$ to the power of $2$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 15 x^{4} \cdot 25 + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.2.3

Multiply $25$ by $15$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.2.4

Raise $- 5$ to the power of $3$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} + 20 x^{3} \cdot - 125 + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.2.5

Multiply $- 125$ by $20$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.2.6

Raise $- 5$ to the power of $4$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 15 x^{2} \cdot 625 + 6 x {(- 5)}^{5} + {(- 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.2.7

Multiply $625$ by $15$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 6 x {(- 5)}^{5} + {(- 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.2.8

Raise $- 5$ to the power of $5$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 6 x \cdot - 3125 + {(- 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.2.9

Multiply $- 3125$ by $6$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + {(- 5)}^{6} + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.2.10

Raise $- 5$ to the power of $6$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 6 \cdot 2) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.3

Multiply $6$ by $2$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 12) {(x - 5)}^{5}) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.4

Use the Binomial Theorem.

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 12) (x^{5} + 5 x^{4} \cdot - 5 + 10 x^{3} {(- 5)}^{2} + 10 x^{2} {(- 5)}^{3} + 5 x {(- 5)}^{4} + {(- 5)}^{5})) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.5

Simplify each term.

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Step 3.14.2.1.5.1

Multiply $- 5$ by $5$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 12) (x^{5} - 25 x^{4} + 10 x^{3} {(- 5)}^{2} + 10 x^{2} {(- 5)}^{3} + 5 x {(- 5)}^{4} + {(- 5)}^{5})) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.5.2

Raise $- 5$ to the power of $2$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 12) (x^{5} - 25 x^{4} + 10 x^{3} \cdot 25 + 10 x^{2} {(- 5)}^{3} + 5 x {(- 5)}^{4} + {(- 5)}^{5})) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.5.3

Multiply $25$ by $10$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 12) (x^{5} - 25 x^{4} + 250 x^{3} + 10 x^{2} {(- 5)}^{3} + 5 x {(- 5)}^{4} + {(- 5)}^{5})) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.5.4

Raise $- 5$ to the power of $3$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 12) (x^{5} - 25 x^{4} + 250 x^{3} + 10 x^{2} \cdot - 125 + 5 x {(- 5)}^{4} + {(- 5)}^{5})) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.5.5

Multiply $- 125$ by $10$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 12) (x^{5} - 25 x^{4} + 250 x^{3} - 1250 x^{2} + 5 x {(- 5)}^{4} + {(- 5)}^{5})) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.5.6

Raise $- 5$ to the power of $4$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 12) (x^{5} - 25 x^{4} + 250 x^{3} - 1250 x^{2} + 5 x \cdot 625 + {(- 5)}^{5})) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.5.7

Multiply $625$ by $5$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 12) (x^{5} - 25 x^{4} + 250 x^{3} - 1250 x^{2} + 3125 x + {(- 5)}^{5})) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.5.8

Raise $- 5$ to the power of $5$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + (6 x + 12) (x^{5} - 25 x^{4} + 250 x^{3} - 1250 x^{2} + 3125 x - 3125)) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.6

Expand $(6 x + 12) (x^{5} - 25 x^{4} + 250 x^{3} - 1250 x^{2} + 3125 x - 3125)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x \cdot x^{5} + 6 x (- 25 x^{4}) + 6 x (250 x^{3}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7

Simplify each term.

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Step 3.14.2.1.7.1

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 3.14.2.1.7.1.1

Move $x^{5}$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 (x^{5} x) + 6 x (- 25 x^{4}) + 6 x (250 x^{3}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.1.2

Multiply $x^{5}$ by $x$ .

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Step 3.14.2.1.7.1.2.1

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

Step 3.14.2.1.7.1.2.2

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

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{5 + 1} + 6 x (- 25 x^{4}) + 6 x (250 x^{3}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.1.3

Add $5$ and $1$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} + 6 x (- 25 x^{4}) + 6 x (250 x^{3}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} + 6 \cdot (- 25 x \cdot x^{4}) + 6 x (250 x^{3}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.3

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

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Step 3.14.2.1.7.3.1

Move $x^{4}$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} + 6 \cdot (- 25 (x^{4} x)) + 6 x (250 x^{3}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.3.2

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

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Step 3.14.2.1.7.3.2.1

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

Step 3.14.2.1.7.3.2.2

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

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} + 6 \cdot (- 25 x^{4 + 1}) + 6 x (250 x^{3}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.3.3

Add $4$ and $1$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} + 6 \cdot (- 25 x^{5}) + 6 x (250 x^{3}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.4

Multiply $6$ by $- 25$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 6 x (250 x^{3}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 6 \cdot (250 x \cdot x^{3}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.6

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

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Step 3.14.2.1.7.6.1

Move $x^{3}$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 6 \cdot (250 (x^{3} x)) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.6.2

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

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Step 3.14.2.1.7.6.2.1

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

Step 3.14.2.1.7.6.2.2

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

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 6 \cdot (250 x^{3 + 1}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.6.3

Add $3$ and $1$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 6 \cdot (250 x^{4}) + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.7

Multiply $6$ by $250$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} + 6 x (- 1250 x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.8

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} + 6 \cdot (- 1250 x \cdot x^{2}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.9

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

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Step 3.14.2.1.7.9.1

Move $x^{2}$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} + 6 \cdot (- 1250 (x^{2} x)) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.9.2

Multiply $x^{2}$ by $x$ .

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Step 3.14.2.1.7.9.2.1

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

Step 3.14.2.1.7.9.2.2

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

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} + 6 \cdot (- 1250 x^{2 + 1}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.9.3

Add $2$ and $1$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} + 6 \cdot (- 1250 x^{3}) + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.10

Multiply $6$ by $- 1250$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 6 x (3125 x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.11

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 6 \cdot (3125 x \cdot x) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.12

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

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Step 3.14.2.1.7.12.1

Move $x$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 6 \cdot (3125 (x \cdot x)) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.12.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 6 \cdot (3125 x^{2}) + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.13

Multiply $6$ by $3125$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 18750 x^{2} + 6 x \cdot - 3125 + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.14

Multiply $- 3125$ by $6$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 18750 x^{2} - 18750 x + 12 x^{5} + 12 (- 25 x^{4}) + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.15

Multiply $- 25$ by $12$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 18750 x^{2} - 18750 x + 12 x^{5} - 300 x^{4} + 12 (250 x^{3}) + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.16

Multiply $250$ by $12$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 18750 x^{2} - 18750 x + 12 x^{5} - 300 x^{4} + 3000 x^{3} + 12 (- 1250 x^{2}) + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.17

Multiply $- 1250$ by $12$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 18750 x^{2} - 18750 x + 12 x^{5} - 300 x^{4} + 3000 x^{3} - 15000 x^{2} + 12 (3125 x) + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.18

Multiply $3125$ by $12$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 18750 x^{2} - 18750 x + 12 x^{5} - 300 x^{4} + 3000 x^{3} - 15000 x^{2} + 37500 x + 12 \cdot - 3125) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.7.19

Multiply $12$ by $- 3125$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 150 x^{5} + 1500 x^{4} - 7500 x^{3} + 18750 x^{2} - 18750 x + 12 x^{5} - 300 x^{4} + 3000 x^{3} - 15000 x^{2} + 37500 x - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.8

Add $- 150 x^{5}$ and $12 x^{5}$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 138 x^{5} + 1500 x^{4} - 7500 x^{3} + 18750 x^{2} - 18750 x - 300 x^{4} + 3000 x^{3} - 15000 x^{2} + 37500 x - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.9

Subtract $300 x^{4}$ from $1500 x^{4}$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 138 x^{5} + 1200 x^{4} - 7500 x^{3} + 18750 x^{2} - 18750 x + 3000 x^{3} - 15000 x^{2} + 37500 x - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.10

Add $- 7500 x^{3}$ and $3000 x^{3}$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 138 x^{5} + 1200 x^{4} - 4500 x^{3} + 18750 x^{2} - 18750 x - 15000 x^{2} + 37500 x - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.11

Subtract $15000 x^{2}$ from $18750 x^{2}$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 138 x^{5} + 1200 x^{4} - 4500 x^{3} + 3750 x^{2} - 18750 x + 37500 x - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.1.12

Add $- 18750 x$ and $37500 x$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 138 x^{5} + 1200 x^{4} - 4500 x^{3} + 3750 x^{2} + 18750 x - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.2

Combine the opposite terms in $x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625 + 6 x^{6} - 138 x^{5} + 1200 x^{4} - 4500 x^{3} + 3750 x^{2} + 18750 x - 37500$ .

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Step 3.14.2.2.1

Add $- 18750 x$ and $18750 x$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 15625 + 6 x^{6} - 138 x^{5} + 1200 x^{4} - 4500 x^{3} + 3750 x^{2} + 0 - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.2.2

Add $x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 15625 + 6 x^{6} - 138 x^{5} + 1200 x^{4} - 4500 x^{3} + 3750 x^{2}$ and $0$ .

$f'' (x) = \frac{(x - 4) (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 15625 + 6 x^{6} - 138 x^{5} + 1200 x^{4} - 4500 x^{3} + 3750 x^{2} - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.3

Add $x^{6}$ and $6 x^{6}$ .

$f'' (x) = \frac{(x - 4) (7 x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 15625 - 138 x^{5} + 1200 x^{4} - 4500 x^{3} + 3750 x^{2} - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.4

Subtract $138 x^{5}$ from $- 30 x^{5}$ .

$f'' (x) = \frac{(x - 4) (7 x^{6} - 168 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 15625 + 1200 x^{4} - 4500 x^{3} + 3750 x^{2} - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.5

Add $375 x^{4}$ and $1200 x^{4}$ .

$f'' (x) = \frac{(x - 4) (7 x^{6} - 168 x^{5} + 1575 x^{4} - 2500 x^{3} + 9375 x^{2} + 15625 - 4500 x^{3} + 3750 x^{2} - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.6

Subtract $4500 x^{3}$ from $- 2500 x^{3}$ .

$f'' (x) = \frac{(x - 4) (7 x^{6} - 168 x^{5} + 1575 x^{4} - 7000 x^{3} + 9375 x^{2} + 15625 + 3750 x^{2} - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.7

Add $9375 x^{2}$ and $3750 x^{2}$ .

$f'' (x) = \frac{(x - 4) (7 x^{6} - 168 x^{5} + 1575 x^{4} - 7000 x^{3} + 13125 x^{2} + 15625 - 37500) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.8

Subtract $37500$ from $15625$ .

$f'' (x) = \frac{(x - 4) (7 x^{6} - 168 x^{5} + 1575 x^{4} - 7000 x^{3} + 13125 x^{2} - 21875) - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.9

Expand $(x - 4) (7 x^{6} - 168 x^{5} + 1575 x^{4} - 7000 x^{3} + 13125 x^{2} - 21875)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{x (7 x^{6}) + x (- 168 x^{5}) + x (1575 x^{4}) + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10

Simplify each term.

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Step 3.14.2.10.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{7 x \cdot x^{6} + x (- 168 x^{5}) + x (1575 x^{4}) + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.2

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 3.14.2.10.2.1

Move $x^{6}$ .

$f'' (x) = \frac{7 (x^{6} x) + x (- 168 x^{5}) + x (1575 x^{4}) + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.2.2

Multiply $x^{6}$ by $x$ .

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Step 3.14.2.10.2.2.1

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

Step 3.14.2.10.2.2.2

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

$f'' (x) = \frac{7 x^{6 + 1} + x (- 168 x^{5}) + x (1575 x^{4}) + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.2.3

Add $6$ and $1$ .

$f'' (x) = \frac{7 x^{7} + x (- 168 x^{5}) + x (1575 x^{4}) + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{7 x^{7} - 168 x \cdot x^{5} + x (1575 x^{4}) + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.4

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 3.14.2.10.4.1

Move $x^{5}$ .

$f'' (x) = \frac{7 x^{7} - 168 (x^{5} x) + x (1575 x^{4}) + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.4.2

Multiply $x^{5}$ by $x$ .

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Step 3.14.2.10.4.2.1

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

Step 3.14.2.10.4.2.2

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

$f'' (x) = \frac{7 x^{7} - 168 x^{5 + 1} + x (1575 x^{4}) + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.4.3

Add $5$ and $1$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + x (1575 x^{4}) + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x \cdot x^{4} + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.6

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

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Step 3.14.2.10.6.1

Move $x^{4}$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 (x^{4} x) + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.6.2

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

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Step 3.14.2.10.6.2.1

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

Step 3.14.2.10.6.2.2

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

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{4 + 1} + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.6.3

Add $4$ and $1$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} + x (- 7000 x^{3}) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.7

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x \cdot x^{3} + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.8

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

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Step 3.14.2.10.8.1

Move $x^{3}$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 (x^{3} x) + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.8.2

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

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Step 3.14.2.10.8.2.1

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

Step 3.14.2.10.8.2.2

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

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{3 + 1} + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.8.3

Add $3$ and $1$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + x (13125 x^{2}) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.9

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x \cdot x^{2} + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.10

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

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Step 3.14.2.10.10.1

Move $x^{2}$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 (x^{2} x) + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.10.2

Multiply $x^{2}$ by $x$ .

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Step 3.14.2.10.10.2.1

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

Step 3.14.2.10.10.2.2

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

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x^{2 + 1} + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.10.3

Add $2$ and $1$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x^{3} + x \cdot - 21875 - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.11

Move $- 21875$ to the left of $x$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x^{3} - 21875 \cdot x - 4 (7 x^{6}) - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.12

Multiply $7$ by $- 4$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x^{3} - 21875 x - 28 x^{6} - 4 (- 168 x^{5}) - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.13

Multiply $- 168$ by $- 4$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x^{3} - 21875 x - 28 x^{6} + 672 x^{5} - 4 (1575 x^{4}) - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.14

Multiply $1575$ by $- 4$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x^{3} - 21875 x - 28 x^{6} + 672 x^{5} - 6300 x^{4} - 4 (- 7000 x^{3}) - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.15

Multiply $- 7000$ by $- 4$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x^{3} - 21875 x - 28 x^{6} + 672 x^{5} - 6300 x^{4} + 28000 x^{3} - 4 (13125 x^{2}) - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.16

Multiply $13125$ by $- 4$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x^{3} - 21875 x - 28 x^{6} + 672 x^{5} - 6300 x^{4} + 28000 x^{3} - 52500 x^{2} - 4 \cdot - 21875 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.10.17

Multiply $- 4$ by $- 21875$ .

$f'' (x) = \frac{7 x^{7} - 168 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x^{3} - 21875 x - 28 x^{6} + 672 x^{5} - 6300 x^{4} + 28000 x^{3} - 52500 x^{2} + 87500 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.11

Subtract $28 x^{6}$ from $- 168 x^{6}$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 1575 x^{5} - 7000 x^{4} + 13125 x^{3} - 21875 x + 672 x^{5} - 6300 x^{4} + 28000 x^{3} - 52500 x^{2} + 87500 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.12

Add $1575 x^{5}$ and $672 x^{5}$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 7000 x^{4} + 13125 x^{3} - 21875 x - 6300 x^{4} + 28000 x^{3} - 52500 x^{2} + 87500 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.13

Subtract $6300 x^{4}$ from $- 7000 x^{4}$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 13125 x^{3} - 21875 x + 28000 x^{3} - 52500 x^{2} + 87500 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.14

Add $13125 x^{3}$ and $28000 x^{3}$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.15

Use the Binomial Theorem.

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} + 6 x^{5} \cdot - 5 + 15 x^{4} {(- 5)}^{2} + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.16

Simplify each term.

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Step 3.14.2.16.1

Multiply $- 5$ by $6$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} - 30 x^{5} + 15 x^{4} {(- 5)}^{2} + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.16.2

Raise $- 5$ to the power of $2$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} - 30 x^{5} + 15 x^{4} \cdot 25 + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.16.3

Multiply $25$ by $15$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} - 30 x^{5} + 375 x^{4} + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.16.4

Raise $- 5$ to the power of $3$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} - 30 x^{5} + 375 x^{4} + 20 x^{3} \cdot - 125 + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.16.5

Multiply $- 125$ by $20$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.16.6

Raise $- 5$ to the power of $4$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 15 x^{2} \cdot 625 + 6 x {(- 5)}^{5} + {(- 5)}^{6}) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.16.7

Multiply $625$ by $15$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.16.8

Raise $- 5$ to the power of $5$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 6 x \cdot - 3125 + {(- 5)}^{6}) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.16.9

Multiply $- 3125$ by $6$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + {(- 5)}^{6}) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.16.10

Raise $- 5$ to the power of $6$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.17

Apply the distributive property.

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + (- 7 x^{6} - 7 (- 30 x^{5}) - 7 (375 x^{4}) - 7 (- 2500 x^{3}) - 7 (9375 x^{2}) - 7 (- 18750 x) - 7 \cdot 15625) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.18

Simplify.

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Step 3.14.2.18.1

Multiply $- 30$ by $- 7$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + (- 7 x^{6} + 210 x^{5} - 7 (375 x^{4}) - 7 (- 2500 x^{3}) - 7 (9375 x^{2}) - 7 (- 18750 x) - 7 \cdot 15625) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.18.2

Multiply $375$ by $- 7$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + (- 7 x^{6} + 210 x^{5} - 2625 x^{4} - 7 (- 2500 x^{3}) - 7 (9375 x^{2}) - 7 (- 18750 x) - 7 \cdot 15625) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.18.3

Multiply $- 2500$ by $- 7$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + (- 7 x^{6} + 210 x^{5} - 2625 x^{4} + 17500 x^{3} - 7 (9375 x^{2}) - 7 (- 18750 x) - 7 \cdot 15625) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.18.4

Multiply $9375$ by $- 7$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + (- 7 x^{6} + 210 x^{5} - 2625 x^{4} + 17500 x^{3} - 65625 x^{2} - 7 (- 18750 x) - 7 \cdot 15625) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.18.5

Multiply $- 18750$ by $- 7$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + (- 7 x^{6} + 210 x^{5} - 2625 x^{4} + 17500 x^{3} - 65625 x^{2} + 131250 x - 7 \cdot 15625) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.18.6

Multiply $- 7$ by $15625$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + (- 7 x^{6} + 210 x^{5} - 2625 x^{4} + 17500 x^{3} - 65625 x^{2} + 131250 x - 109375) (x + 2)}{{(x - 4)}^{8}}$

Step 3.14.2.19

Expand $(- 7 x^{6} + 210 x^{5} - 2625 x^{4} + 17500 x^{3} - 65625 x^{2} + 131250 x - 109375) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{6} x - 7 x^{6} \cdot 2 + 210 x^{5} x + 210 x^{5} \cdot 2 - 2625 x^{4} x - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20

Simplify each term.

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Step 3.14.2.20.1

Multiply $x^{6}$ by $x$ by adding the exponents.

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Step 3.14.2.20.1.1

Move $x$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 (x \cdot x^{6}) - 7 x^{6} \cdot 2 + 210 x^{5} x + 210 x^{5} \cdot 2 - 2625 x^{4} x - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.1.2

Multiply $x$ by $x^{6}$ .

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Step 3.14.2.20.1.2.1

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

Step 3.14.2.20.1.2.2

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

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{1 + 6} - 7 x^{6} \cdot 2 + 210 x^{5} x + 210 x^{5} \cdot 2 - 2625 x^{4} x - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.1.3

Add $1$ and $6$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 7 x^{6} \cdot 2 + 210 x^{5} x + 210 x^{5} \cdot 2 - 2625 x^{4} x - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.2

Multiply $2$ by $- 7$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{5} x + 210 x^{5} \cdot 2 - 2625 x^{4} x - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.3

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 3.14.2.20.3.1

Move $x$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 (x \cdot x^{5}) + 210 x^{5} \cdot 2 - 2625 x^{4} x - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.3.2

Multiply $x$ by $x^{5}$ .

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Step 3.14.2.20.3.2.1

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

Step 3.14.2.20.3.2.2

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

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{1 + 5} + 210 x^{5} \cdot 2 - 2625 x^{4} x - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.3.3

Add $1$ and $5$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 210 x^{5} \cdot 2 - 2625 x^{4} x - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.4

Multiply $2$ by $210$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{4} x - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.5

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

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Step 3.14.2.20.5.1

Move $x$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 (x \cdot x^{4}) - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.5.2

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

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Step 3.14.2.20.5.2.1

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

Step 3.14.2.20.5.2.2

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

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{1 + 4} - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.5.3

Add $1$ and $4$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 2625 x^{4} \cdot 2 + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.6

Multiply $2$ by $- 2625$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{3} x + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.7

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

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Step 3.14.2.20.7.1

Move $x$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 (x \cdot x^{3}) + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.7.2

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

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Step 3.14.2.20.7.2.1

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

Step 3.14.2.20.7.2.2

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

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{1 + 3} + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.7.3

Add $1$ and $3$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 17500 x^{3} \cdot 2 - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.8

Multiply $2$ by $17500$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{2} x - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.9

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

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Step 3.14.2.20.9.1

Move $x$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 (x \cdot x^{2}) - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.9.2

Multiply $x$ by $x^{2}$ .

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Step 3.14.2.20.9.2.1

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

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 (x \cdot x^{2}) - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.9.2.2

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

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{1 + 2} - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.9.3

Add $1$ and $2$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} - 65625 x^{2} \cdot 2 + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.10

Multiply $2$ by $- 65625$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} - 131250 x^{2} + 131250 x \cdot x + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.11

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

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Step 3.14.2.20.11.1

Move $x$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} - 131250 x^{2} + 131250 (x \cdot x) + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.11.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} - 131250 x^{2} + 131250 x^{2} + 131250 x \cdot 2 - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.12

Multiply $2$ by $131250$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} - 131250 x^{2} + 131250 x^{2} + 262500 x - 109375 x - 109375 \cdot 2}{{(x - 4)}^{8}}$

Step 3.14.2.20.13

Multiply $- 109375$ by $2$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} - 131250 x^{2} + 131250 x^{2} + 262500 x - 109375 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.21

Combine the opposite terms in $- 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} - 131250 x^{2} + 131250 x^{2} + 262500 x - 109375 x - 218750$ .

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Step 3.14.2.21.1

Add $- 131250 x^{2}$ and $131250 x^{2}$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} + 0 + 262500 x - 109375 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.21.2

Add $- 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3}$ and $0$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} - 14 x^{6} + 210 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} + 262500 x - 109375 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.22

Add $- 14 x^{6}$ and $210 x^{6}$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} + 196 x^{6} + 420 x^{5} - 2625 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} + 262500 x - 109375 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.23

Subtract $2625 x^{5}$ from $420 x^{5}$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} + 196 x^{6} - 2205 x^{5} - 5250 x^{4} + 17500 x^{4} + 35000 x^{3} - 65625 x^{3} + 262500 x - 109375 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.24

Add $- 5250 x^{4}$ and $17500 x^{4}$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} + 196 x^{6} - 2205 x^{5} + 12250 x^{4} + 35000 x^{3} - 65625 x^{3} + 262500 x - 109375 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.25

Subtract $65625 x^{3}$ from $35000 x^{3}$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} + 196 x^{6} - 2205 x^{5} + 12250 x^{4} - 30625 x^{3} + 262500 x - 109375 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.26

Subtract $109375 x$ from $262500 x$ .

$f'' (x) = \frac{7 x^{7} - 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 7 x^{7} + 196 x^{6} - 2205 x^{5} + 12250 x^{4} - 30625 x^{3} + 153125 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.27

Subtract $7 x^{7}$ from $7 x^{7}$ .

$f'' (x) = \frac{- 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + 0 + 196 x^{6} - 2205 x^{5} + 12250 x^{4} - 30625 x^{3} + 153125 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.28

Add $- 196 x^{6}$ and $0$ .

$f'' (x) = \frac{- 196 x^{6} + 2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + 196 x^{6} - 2205 x^{5} + 12250 x^{4} - 30625 x^{3} + 153125 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.29

Add $- 196 x^{6}$ and $196 x^{6}$ .

$f'' (x) = \frac{2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + 0 - 2205 x^{5} + 12250 x^{4} - 30625 x^{3} + 153125 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.30

Add $2247 x^{5}$ and $0$ .

$f'' (x) = \frac{2247 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 2205 x^{5} + 12250 x^{4} - 30625 x^{3} + 153125 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.31

Subtract $2205 x^{5}$ from $2247 x^{5}$ .

$f'' (x) = \frac{42 x^{5} - 13300 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 + 12250 x^{4} - 30625 x^{3} + 153125 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.32

Add $- 13300 x^{4}$ and $12250 x^{4}$ .

$f'' (x) = \frac{42 x^{5} - 1050 x^{4} + 41125 x^{3} - 21875 x - 52500 x^{2} + 87500 - 30625 x^{3} + 153125 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.33

Subtract $30625 x^{3}$ from $41125 x^{3}$ .

$f'' (x) = \frac{42 x^{5} - 1050 x^{4} + 10500 x^{3} - 21875 x - 52500 x^{2} + 87500 + 153125 x - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.34

Add $- 21875 x$ and $153125 x$ .

$f'' (x) = \frac{42 x^{5} - 1050 x^{4} + 10500 x^{3} + 131250 x - 52500 x^{2} + 87500 - 218750}{{(x - 4)}^{8}}$

Step 3.14.2.35

Subtract $218750$ from $87500$ .

$f'' (x) = \frac{42 x^{5} - 1050 x^{4} + 10500 x^{3} + 131250 x - 52500 x^{2} - 131250}{{(x - 4)}^{8}}$

Step 3.14.2.36

Reorder terms.

$f'' (x) = \frac{42 x^{5} - 1050 x^{4} + 10500 x^{3} - 52500 x^{2} + 131250 x - 131250}{{(x - 4)}^{8}}$

Step 3.14.2.37

Rewrite $42 x^{5} - 1050 x^{4} + 10500 x^{3} - 52500 x^{2} + 131250 x - 131250$ in a factored form.

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Step 3.14.2.37.1

Factor $42$ out of $42 x^{5} - 1050 x^{4} + 10500 x^{3} - 52500 x^{2} + 131250 x - 131250$ .

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Step 3.14.2.37.1.1

Factor $42$ out of $42 x^{5}$ .

$f'' (x) = \frac{42 (x^{5}) - 1050 x^{4} + 10500 x^{3} - 52500 x^{2} + 131250 x - 131250}{{(x - 4)}^{8}}$

Step 3.14.2.37.1.2

Factor $42$ out of $- 1050 x^{4}$ .

$f'' (x) = \frac{42 (x^{5}) + 42 (- 25 x^{4}) + 10500 x^{3} - 52500 x^{2} + 131250 x - 131250}{{(x - 4)}^{8}}$

Step 3.14.2.37.1.3

Factor $42$ out of $10500 x^{3}$ .

$f'' (x) = \frac{42 (x^{5}) + 42 (- 25 x^{4}) + 42 (250 x^{3}) - 52500 x^{2} + 131250 x - 131250}{{(x - 4)}^{8}}$

Step 3.14.2.37.1.4

Factor $42$ out of $- 52500 x^{2}$ .

$f'' (x) = \frac{42 (x^{5}) + 42 (- 25 x^{4}) + 42 (250 x^{3}) + 42 (- 1250 x^{2}) + 131250 x - 131250}{{(x - 4)}^{8}}$

Step 3.14.2.37.1.5

Factor $42$ out of $131250 x$ .

$f'' (x) = \frac{42 (x^{5}) + 42 (- 25 x^{4}) + 42 (250 x^{3}) + 42 (- 1250 x^{2}) + 42 (3125 x) - 131250}{{(x - 4)}^{8}}$

Step 3.14.2.37.1.6

Factor $42$ out of $- 131250$ .

$f'' (x) = \frac{42 x^{5} + 42 (- 25 x^{4}) + 42 (250 x^{3}) + 42 (- 1250 x^{2}) + 42 (3125 x) + 42 \cdot - 3125}{{(x - 4)}^{8}}$

Step 3.14.2.37.1.7

Factor $42$ out of $42 x^{5} + 42 (- 25 x^{4})$ .

$f'' (x) = \frac{42 (x^{5} - 25 x^{4}) + 42 (250 x^{3}) + 42 (- 1250 x^{2}) + 42 (3125 x) + 42 \cdot - 3125}{{(x - 4)}^{8}}$

Step 3.14.2.37.1.8

Factor $42$ out of $42 (x^{5} - 25 x^{4}) + 42 (250 x^{3})$ .

$f'' (x) = \frac{42 (x^{5} - 25 x^{4} + 250 x^{3}) + 42 (- 1250 x^{2}) + 42 (3125 x) + 42 \cdot - 3125}{{(x - 4)}^{8}}$

Step 3.14.2.37.1.9

Factor $42$ out of $42 (x^{5} - 25 x^{4} + 250 x^{3}) + 42 (- 1250 x^{2})$ .

$f'' (x) = \frac{42 (x^{5} - 25 x^{4} + 250 x^{3} - 1250 x^{2}) + 42 (3125 x) + 42 \cdot - 3125}{{(x - 4)}^{8}}$

Step 3.14.2.37.1.10

Factor $42$ out of $42 (x^{5} - 25 x^{4} + 250 x^{3} - 1250 x^{2}) + 42 (3125 x)$ .

$f'' (x) = \frac{42 (x^{5} - 25 x^{4} + 250 x^{3} - 1250 x^{2} + 3125 x) + 42 \cdot - 3125}{{(x - 4)}^{8}}$

Step 3.14.2.37.1.11

Factor $42$ out of $42 (x^{5} - 25 x^{4} + 250 x^{3} - 1250 x^{2} + 3125 x) + 42 \cdot - 3125$ .

$f'' (x) = \frac{42 (x^{5} - 25 x^{4} + 250 x^{3} - 1250 x^{2} + 3125 x - 3125)}{{(x - 4)}^{8}}$

Step 3.14.2.37.2

Make each term match the terms from the binomial theorem formula.

$f'' (x) = \frac{42 (x^{5} - 5 \cdot (5 x^{4}) + 10 \cdot (5^{2} x^{3}) - 10 \cdot (5^{3} x^{2}) + 5 \cdot (5^{4} x) - 1 \cdot 5^{5})}{{(x - 4)}^{8}}$

Step 3.14.2.37.3

Factor $x^{5} - 5 \cdot 5 x^{4} + 10 \cdot 5^{2} x^{3} - 10 \cdot 5^{3} x^{2} + 5 \cdot 5^{4} x - 1 \cdot 5^{5}$ using the binomial theorem.

$f'' (x) = \frac{42 {(x - 5)}^{5}}{{(x - 4)}^{8}}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

$\frac{{(x - 4)}^{6} \frac{d}{d x} [{(x - 5)}^{7}] - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{({(x - 4)}^{6})}^{2}}$

Step 5.1.2

Multiply the exponents in ${({(x - 4)}^{6})}^{2}$ .

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Step 5.1.2.1

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

$\frac{{(x - 4)}^{6} \frac{d}{d x} [{(x - 5)}^{7}] - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{6 \cdot 2}}$

Step 5.1.2.2

Multiply $6$ by $2$ .

$\frac{{(x - 4)}^{6} \frac{d}{d x} [{(x - 5)}^{7}] - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 5.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = x - 5$ .

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Step 5.1.3.1

To apply the Chain Rule, set $u_{1}$ as $x - 5$ .

$\frac{{(x - 4)}^{6} (\frac{d}{d u_{1}} [{u_{1}}^{7}] \frac{d}{d x} [x - 5]) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 5.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 7$ .

$\frac{{(x - 4)}^{6} (7 {u_{1}}^{6} \frac{d}{d x} [x - 5]) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 5.1.3.3

Replace all occurrences of $u_{1}$ with $x - 5$ .

$\frac{{(x - 4)}^{6} (7 {(x - 5)}^{6} \frac{d}{d x} [x - 5]) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 5.1.4

Differentiate.

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Step 5.1.4.1

Move $7$ to the left of ${(x - 4)}^{6}$ .

$\frac{7 \cdot {(x - 4)}^{6} {(x - 5)}^{6} \frac{d}{d x} [x - 5] - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 5.1.4.2

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 5.1.4.3

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} (1 + \frac{d}{d x} [- 5]) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 5.1.4.4

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} (1 + 0) - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 5.1.4.5

Simplify the expression.

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Step 5.1.4.5.1

Add $1$ and $0$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} \cdot 1 - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 5.1.4.5.2

Multiply $7$ by $1$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - {(x - 5)}^{7} \frac{d}{d x} [{(x - 4)}^{6}]}{{(x - 4)}^{12}}$

Step 5.1.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = x - 4$ .

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Step 5.1.5.1

To apply the Chain Rule, set $u_{2}$ as $x - 4$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - {(x - 5)}^{7} (\frac{d}{d u_{2}} [{u_{2}}^{6}] \frac{d}{d x} [x - 4])}{{(x - 4)}^{12}}$

Step 5.1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 6$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - {(x - 5)}^{7} (6 {u_{2}}^{5} \frac{d}{d x} [x - 4])}{{(x - 4)}^{12}}$

Step 5.1.5.3

Replace all occurrences of $u_{2}$ with $x - 4$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - {(x - 5)}^{7} (6 {(x - 4)}^{5} \frac{d}{d x} [x - 4])}{{(x - 4)}^{12}}$

Step 5.1.6

Differentiate.

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Step 5.1.6.1

Multiply $6$ by $- 1$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} ({(x - 4)}^{5} \frac{d}{d x} [x - 4])}{{(x - 4)}^{12}}$

Step 5.1.6.2

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} ({(x - 4)}^{5} (\frac{d}{d x} [x] + \frac{d}{d x} [- 4]))}{{(x - 4)}^{12}}$

Step 5.1.6.3

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} ({(x - 4)}^{5} (1 + \frac{d}{d x} [- 4]))}{{(x - 4)}^{12}}$

Step 5.1.6.4

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

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} ({(x - 4)}^{5} (1 + 0))}{{(x - 4)}^{12}}$

Step 5.1.6.5

Simplify the expression.

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Step 5.1.6.5.1

Add $1$ and $0$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} ({(x - 4)}^{5} \cdot 1)}{{(x - 4)}^{12}}$

Step 5.1.6.5.2

Multiply ${(x - 4)}^{5}$ by $1$ .

$\frac{7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} {(x - 4)}^{5}}{{(x - 4)}^{12}}$

Step 5.1.7

Simplify.

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Step 5.1.7.1

Simplify the numerator.

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Step 5.1.7.1.1

Factor ${(x - 4)}^{5} {(x - 5)}^{6}$ out of $7 {(x - 4)}^{6} {(x - 5)}^{6} - 6 {(x - 5)}^{7} {(x - 4)}^{5}$ .

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Step 5.1.7.1.1.1

Factor ${(x - 4)}^{5} {(x - 5)}^{6}$ out of $7 {(x - 4)}^{6} {(x - 5)}^{6}$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 (x - 4)) - 6 {(x - 5)}^{7} {(x - 4)}^{5}}{{(x - 4)}^{12}}$

Step 5.1.7.1.1.2

Factor ${(x - 4)}^{5} {(x - 5)}^{6}$ out of $- 6 {(x - 5)}^{7} {(x - 4)}^{5}$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 (x - 4)) + {(x - 4)}^{5} {(x - 5)}^{6} (- 6 (x - 5))}{{(x - 4)}^{12}}$

Step 5.1.7.1.1.3

Factor ${(x - 4)}^{5} {(x - 5)}^{6}$ out of ${(x - 4)}^{5} {(x - 5)}^{6} (7 (x - 4)) + {(x - 4)}^{5} {(x - 5)}^{6} (- 6 (x - 5))$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 (x - 4) - 6 (x - 5))}{{(x - 4)}^{12}}$

Step 5.1.7.1.2

Apply the distributive property.

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 x + 7 \cdot - 4 - 6 (x - 5))}{{(x - 4)}^{12}}$

Step 5.1.7.1.3

Multiply $7$ by $- 4$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 x - 28 - 6 (x - 5))}{{(x - 4)}^{12}}$

Step 5.1.7.1.4

Apply the distributive property.

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 x - 28 - 6 x - 6 \cdot - 5)}{{(x - 4)}^{12}}$

Step 5.1.7.1.5

Multiply $- 6$ by $- 5$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (7 x - 28 - 6 x + 30)}{{(x - 4)}^{12}}$

Step 5.1.7.1.6

Subtract $6 x$ from $7 x$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (x - 28 + 30)}{{(x - 4)}^{12}}$

Step 5.1.7.1.7

Add $- 28$ and $30$ .

$\frac{{(x - 4)}^{5} {(x - 5)}^{6} (x + 2)}{{(x - 4)}^{12}}$

Step 5.1.7.2

Cancel the common factor of ${(x - 4)}^{5}$ and ${(x - 4)}^{12}$ .

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Step 5.1.7.2.1

Factor ${(x - 4)}^{5}$ out of ${(x - 4)}^{5} {(x - 5)}^{6} (x + 2)$ .

$\frac{{(x - 4)}^{5} ({(x - 5)}^{6} (x + 2))}{{(x - 4)}^{12}}$

Step 5.1.7.2.2

Cancel the common factors.

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Step 5.1.7.2.2.1

Factor ${(x - 4)}^{5}$ out of ${(x - 4)}^{12}$ .

$\frac{{(x - 4)}^{5} ({(x - 5)}^{6} (x + 2))}{{(x - 4)}^{5} {(x - 4)}^{7}}$

Step 5.1.7.2.2.2

Cancel the common factor.

$\frac{{(x - 4)}^{5} ({(x - 5)}^{6} (x + 2))}{{(x - 4)}^{5} {(x - 4)}^{7}}$

Step 5.1.7.2.2.3

Rewrite the expression.

$f' (x) = \frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}}$ .

$\frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}} = 0$

Step 6.2

Set the numerator equal to zero.

${(x - 5)}^{6} (x + 2) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x - 5)}^{6} = 0$

$x + 2 = 0$

Step 6.3.2

Set ${(x - 5)}^{6}$ equal to $0$ and solve for $x$ .

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Step 6.3.2.1

Set ${(x - 5)}^{6}$ equal to $0$ .

${(x - 5)}^{6} = 0$

Step 6.3.2.2

Solve ${(x - 5)}^{6} = 0$ for $x$ .

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Step 6.3.2.2.1

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 6.3.2.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 6.3.3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 6.3.3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 6.3.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6.3.4

The final solution is all the values that make ${(x - 5)}^{6} (x + 2) = 0$ true.

$x = 5, - 2$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}}$ equal to $0$ to find where the expression is undefined.

${(x - 4)}^{7} = 0$

Step 7.2

Solve for $x$ .

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Step 7.2.1

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 7.2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 8

Critical points to evaluate.

$x = 5, - 2$

Step 9

Evaluate the second derivative at $x = 5$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{42 {((5) - 5)}^{5}}{{((5) - 4)}^{8}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Subtract $5$ from $5$ .

$\frac{42 \cdot 0^{5}}{{(5 - 4)}^{8}}$

Step 10.1.2

Raising $0$ to any positive power yields $0$ .

$\frac{42 \cdot 0}{{(5 - 4)}^{8}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Subtract $4$ from $5$ .

$\frac{42 \cdot 0}{1^{8}}$

Step 10.2.2

One to any power is one.

$\frac{42 \cdot 0}{1}$

Step 10.3

Simplify the expression.

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Step 10.3.1

Multiply $42$ by $0$ .

$\frac{0}{1}$

Step 10.3.2

Divide $0$ by $1$ .

$0$

Step 11

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 11.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - 2) \cup (- 2, 5) \cup (5, \infty)$

Step 11.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, - 2)$ in the first derivative $\frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}}$ to check if the result is negative or positive.

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Step 11.2.1

Replace the variable $x$ with $- 4$ in the expression.

$f' (- 4) = \frac{{((- 4) - 5)}^{6} ((- 4) + 2)}{{((- 4) - 4)}^{7}}$

Step 11.2.2

Simplify the result.

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Step 11.2.2.1

Simplify the numerator.

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Step 11.2.2.1.1

Subtract $5$ from $- 4$ .

$f' (- 4) = \frac{{(- 9)}^{6} (- 4 + 2)}{{(- 4 - 4)}^{7}}$

Step 11.2.2.1.2

Add $- 4$ and $2$ .

$f' (- 4) = \frac{{(- 9)}^{6} \cdot - 2}{{(- 4 - 4)}^{7}}$

Step 11.2.2.1.3

Raise $- 9$ to the power of $6$ .

$f' (- 4) = \frac{531441 \cdot - 2}{{(- 4 - 4)}^{7}}$

Step 11.2.2.2

Simplify the denominator.

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Step 11.2.2.2.1

Subtract $4$ from $- 4$ .

$f' (- 4) = \frac{531441 \cdot - 2}{{(- 8)}^{7}}$

Step 11.2.2.2.2

Raise $- 8$ to the power of $7$ .

$f' (- 4) = \frac{531441 \cdot - 2}{- 2097152}$

Step 11.2.2.3

Reduce the expression by cancelling the common factors.

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Step 11.2.2.3.1

Multiply $531441$ by $- 2$ .

$f' (- 4) = \frac{- 1062882}{- 2097152}$

Step 11.2.2.3.2

Cancel the common factor of $- 1062882$ and $- 2097152$ .

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Step 11.2.2.3.2.1

Factor $- 2$ out of $- 1062882$ .

$f' (- 4) = \frac{- 2 \cdot 531441}{- 2097152}$

Step 11.2.2.3.2.2

Cancel the common factors.

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Step 11.2.2.3.2.2.1

Factor $- 2$ out of $- 2097152$ .

$f' (- 4) = \frac{- 2 \cdot 531441}{- 2 \cdot 1048576}$

Step 11.2.2.3.2.2.2

Cancel the common factor.

$f' (- 4) = \frac{- 2 \cdot 531441}{- 2 \cdot 1048576}$

Step 11.2.2.3.2.2.3

Rewrite the expression.

$f' (- 4) = \frac{531441}{1048576}$

Step 11.2.2.4

The final answer is $\frac{531441}{1048576}$ .

$\frac{531441}{1048576}$

Step 11.3

Substitute any number, such as $0$ , from the interval $(- 2, 5)$ in the first derivative $\frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}}$ to check if the result is negative or positive.

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Step 11.3.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = \frac{{((0) - 5)}^{6} ((0) + 2)}{{((0) - 4)}^{7}}$

Step 11.3.2

Simplify the result.

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Step 11.3.2.1

Simplify the numerator.

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Step 11.3.2.1.1

Subtract $5$ from $0$ .

$f' (0) = \frac{{(- 5)}^{6} (0 + 2)}{{(0 - 4)}^{7}}$

Step 11.3.2.1.2

Add $0$ and $2$ .

$f' (0) = \frac{{(- 5)}^{6} \cdot 2}{{(0 - 4)}^{7}}$

Step 11.3.2.1.3

Raise $- 5$ to the power of $6$ .

$f' (0) = \frac{15625 \cdot 2}{{(0 - 4)}^{7}}$

Step 11.3.2.2

Simplify the denominator.

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Step 11.3.2.2.1

Subtract $4$ from $0$ .

$f' (0) = \frac{15625 \cdot 2}{{(- 4)}^{7}}$

Step 11.3.2.2.2

Raise $- 4$ to the power of $7$ .

$f' (0) = \frac{15625 \cdot 2}{- 16384}$

Step 11.3.2.3

Reduce the expression by cancelling the common factors.

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Step 11.3.2.3.1

Multiply $15625$ by $2$ .

$f' (0) = \frac{31250}{- 16384}$

Step 11.3.2.3.2

Cancel the common factor of $31250$ and $- 16384$ .

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Step 11.3.2.3.2.1

Factor $2$ out of $31250$ .

$f' (0) = \frac{2 (15625)}{- 16384}$

Step 11.3.2.3.2.2

Cancel the common factors.

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Step 11.3.2.3.2.2.1

Factor $2$ out of $- 16384$ .

$f' (0) = \frac{2 \cdot 15625}{2 \cdot - 8192}$

Step 11.3.2.3.2.2.2

Cancel the common factor.

$f' (0) = \frac{2 \cdot 15625}{2 \cdot - 8192}$

Step 11.3.2.3.2.2.3

Rewrite the expression.

$f' (0) = \frac{15625}{- 8192}$

Step 11.3.2.3.3

Move the negative in front of the fraction.

$f' (0) = - \frac{15625}{8192}$

Step 11.3.2.4

The final answer is $- \frac{15625}{8192}$ .

$- \frac{15625}{8192}$

Step 11.4

Substitute any number, such as $8$ , from the interval $(5, \infty)$ in the first derivative $\frac{{(x - 5)}^{6} (x + 2)}{{(x - 4)}^{7}}$ to check if the result is negative or positive.

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Step 11.4.1

Replace the variable $x$ with $8$ in the expression.

$f' (8) = \frac{{((8) - 5)}^{6} ((8) + 2)}{{((8) - 4)}^{7}}$

Step 11.4.2

Simplify the result.

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Step 11.4.2.1

Simplify the numerator.

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Step 11.4.2.1.1

Subtract $5$ from $8$ .

$f' (8) = \frac{3^{6} (8 + 2)}{{(8 - 4)}^{7}}$

Step 11.4.2.1.2

Add $8$ and $2$ .

$f' (8) = \frac{3^{6} \cdot 10}{{(8 - 4)}^{7}}$

Step 11.4.2.1.3

Raise $3$ to the power of $6$ .

$f' (8) = \frac{729 \cdot 10}{{(8 - 4)}^{7}}$

Step 11.4.2.2

Simplify the denominator.

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Step 11.4.2.2.1

Subtract $4$ from $8$ .

$f' (8) = \frac{729 \cdot 10}{4^{7}}$

Step 11.4.2.2.2

Raise $4$ to the power of $7$ .

$f' (8) = \frac{729 \cdot 10}{16384}$

Step 11.4.2.3

Reduce the expression by cancelling the common factors.

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Step 11.4.2.3.1

Multiply $729$ by $10$ .

$f' (8) = \frac{7290}{16384}$

Step 11.4.2.3.2

Cancel the common factor of $7290$ and $16384$ .

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Step 11.4.2.3.2.1

Factor $2$ out of $7290$ .

$f' (8) = \frac{2 (3645)}{16384}$

Step 11.4.2.3.2.2

Cancel the common factors.

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Step 11.4.2.3.2.2.1

Factor $2$ out of $16384$ .

$f' (8) = \frac{2 \cdot 3645}{2 \cdot 8192}$

Step 11.4.2.3.2.2.2

Cancel the common factor.

$f' (8) = \frac{2 \cdot 3645}{2 \cdot 8192}$

Step 11.4.2.3.2.2.3

Rewrite the expression.

$f' (8) = \frac{3645}{8192}$

Step 11.4.2.4

The final answer is $\frac{3645}{8192}$ .

$\frac{3645}{8192}$

Step 11.5

Since the first derivative changed signs from positive to negative around $x = - 2$ , then $x = - 2$ is a local maximum.

$x = - 2$ is a local maximum

Step 11.6

Since the first derivative changed signs from negative to positive around $x = 5$ , then $x = 5$ is a local minimum.

$x = 5$ is a local minimum

Step 11.7

These are the local extrema for $f (x) = \frac{{(x - 5)}^{7}}{{(x - 4)}^{6}}$ .

$x = - 2$ is a local maximum

$x = 5$ is a local minimum

$x = - 2$ is a local maximum

$x = 5$ is a local minimum

Step 12