Calculus Examples

$f (x) = 10 x + 26 \sqrt{1296 + {(79 - x)}^{2}}$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $10 x + 26 \sqrt{1296 + {(79 - x)}^{2}}$ with respect to $x$ is $\frac{d}{d x} [10 x] + \frac{d}{d x} [26 \sqrt{1296 + {(79 - x)}^{2}}]$ .

$\frac{d}{d x} [10 x] + \frac{d}{d x} [26 \sqrt{1296 + {(79 - x)}^{2}}]$

Step 1.2

Evaluate $\frac{d}{d x} [10 x]$ .

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Step 1.2.1

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

$10 \frac{d}{d x} [x] + \frac{d}{d x} [26 \sqrt{1296 + {(79 - x)}^{2}}]$

Step 1.2.2

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

$10 \cdot 1 + \frac{d}{d x} [26 \sqrt{1296 + {(79 - x)}^{2}}]$

Step 1.2.3

Multiply $10$ by $1$ .

$10 + \frac{d}{d x} [26 \sqrt{1296 + {(79 - x)}^{2}}]$

Step 1.3

Evaluate $\frac{d}{d x} [26 \sqrt{1296 + {(79 - x)}^{2}}]$ .

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Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1296 + {(79 - x)}^{2}}$ as ${(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}$ .

$10 + \frac{d}{d x} [26 {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}]$

Step 1.3.2

Since $26$ is constant with respect to $x$ , the derivative of $26 {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}$ with respect to $x$ is $26 \frac{d}{d x} [{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}]$ .

$10 + 26 \frac{d}{d x} [{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}]$

Step 1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 1296 + {(79 - x)}^{2}$ .

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Step 1.3.3.1

To apply the Chain Rule, set $u_{1}$ as $1296 + {(79 - x)}^{2}$ .

$10 + 26 (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [1296 + {(79 - x)}^{2}])$

Step 1.3.3.2

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

$10 + 26 (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [1296 + {(79 - x)}^{2}])$

Step 1.3.3.3

Replace all occurrences of $u_{1}$ with $1296 + {(79 - x)}^{2}$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [1296 + {(79 - x)}^{2}])$

Step 1.3.4

By the Sum Rule, the derivative of $1296 + {(79 - x)}^{2}$ with respect to $x$ is $\frac{d}{d x} [1296] + \frac{d}{d x} [{(79 - x)}^{2}]$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (\frac{d}{d x} [1296] + \frac{d}{d x} [{(79 - x)}^{2}]))$

Step 1.3.5

Since $1296$ is constant with respect to $x$ , the derivative of $1296$ with respect to $x$ is $0$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [{(79 - x)}^{2}]))$

Step 1.3.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 79 - x$ .

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Step 1.3.6.1

To apply the Chain Rule, set $u_{2}$ as $79 - x$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [79 - x]))$

Step 1.3.6.2

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

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 u_{2} \frac{d}{d x} [79 - x]))$

Step 1.3.6.3

Replace all occurrences of $u_{2}$ with $79 - x$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) \frac{d}{d x} [79 - x]))$

Step 1.3.7

By the Sum Rule, the derivative of $79 - x$ with respect to $x$ is $\frac{d}{d x} [79] + \frac{d}{d x} [- x]$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (\frac{d}{d x} [79] + \frac{d}{d x} [- x])))$

Step 1.3.8

Since $79$ is constant with respect to $x$ , the derivative of $79$ with respect to $x$ is $0$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (0 + \frac{d}{d x} [- x])))$

Step 1.3.9

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

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (0 - \frac{d}{d x} [x])))$

Step 1.3.10

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

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (0 - 1 \cdot 1)))$

Step 1.3.11

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

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))$

Step 1.3.12

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

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))$

Step 1.3.13

Combine the numerators over the common denominator.

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1 - 1 \cdot 2}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))$

Step 1.3.14

Simplify the numerator.

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Step 1.3.14.1

Multiply $- 1$ by $2$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{1 - 2}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))$

Step 1.3.14.2

Subtract $2$ from $1$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{\frac{- 1}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))$

Step 1.3.15

Move the negative in front of the fraction.

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))$

Step 1.3.16

Multiply $- 1$ by $1$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}} (0 + 2 (79 - x) (0 - 1)))$

Step 1.3.17

Subtract $1$ from $0$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}} (0 + 2 (79 - x) \cdot - 1))$

Step 1.3.18

Multiply $- 1$ by $2$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}} (0 - 2 (79 - x)))$

Step 1.3.19

Subtract $2 (79 - x)$ from $0$ .

$10 + 26 (\frac{1}{2} {(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}} (- 2 (79 - x)))$

Step 1.3.20

Combine $\frac{1}{2}$ and ${(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}}$ .

$10 + 26 (\frac{{(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}}}{2} (- 2 (79 - x)))$

Step 1.3.21

Combine $- 2$ and $\frac{{(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}}}{2}$ .

$10 + 26 (\frac{- 2 {(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}}}{2} (79 - x))$

Step 1.3.22

Move ${(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$10 + 26 (\frac{- 2}{2 {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} (79 - x))$

Step 1.3.23

Factor $2$ out of $- 2$ .

$10 + 26 (\frac{2 \cdot - 1}{2 {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} (79 - x))$

Step 1.3.24

Cancel the common factors.

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Step 1.3.24.1

Factor $2$ out of $2 {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}$ .

$10 + 26 (\frac{2 \cdot - 1}{2 ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})} (79 - x))$

Step 1.3.24.2

Cancel the common factor.

$10 + 26 (\frac{2 \cdot - 1}{2 {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} (79 - x))$

Step 1.3.24.3

Rewrite the expression.

$10 + 26 (\frac{- 1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} (79 - x))$

Step 1.3.25

Move the negative in front of the fraction.

$10 + 26 (- \frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} (79 - x))$

Step 1.3.26

Multiply $- 1$ by $26$ .

$10 - 26 (\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} (79 - x))$

Step 1.3.27

Combine $\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}$ and $- 26$ .

$10 + \frac{- 26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} (79 - x)$

Step 1.3.28

Move the negative in front of the fraction.

$10 - \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} (79 - x)$

Step 1.4

Reorder terms.

$- (79 - x) \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} + 10$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- (79 - x) \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} + 10$ with respect to $x$ is $\frac{d}{d x} [- (79 - x) \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}] + \frac{d}{d x} [10]$ .

$f'' (x) = \frac{d}{d x} (- (79 - x) \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}) + \frac{d}{d x} (10)$

Step 2.2

Evaluate $\frac{d}{d x} [- (79 - x) \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}]$ .

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Step 2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (79 - x) \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}$ with respect to $x$ is $- \frac{d}{d x} [(79 - x) \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}]$ .

$f'' (x) = - \frac{d}{d x} ((79 - x) (\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}})) + \frac{d}{d x} (10)$

Step 2.2.2

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 $f (x) = 79 - x$ and $g (x) = \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}$ .

$f'' (x) = - ((79 - x) \frac{d}{d x} (\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.3

Since $26$ is constant with respect to $x$ , the derivative of $\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}$ with respect to $x$ is $26 \frac{d}{d x} [\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}]$ .

$f'' (x) = - ((79 - x) (26 \frac{d}{d x} (\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}})) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.4

Rewrite $\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}$ as ${({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 1}$ .

$f'' (x) = - ((79 - x) (26 \frac{d}{d x} ({({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 1})) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}$ .

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Step 2.2.5.1

To apply the Chain Rule, set $u_{1}$ as ${(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}$ .

$f'' (x) = - ((79 - x) (26 (\frac{d}{d u (1)} ({u_{1}}^{- 1}) \frac{d}{d x} ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.5.2

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

$f'' (x) = - ((79 - x) (26 (- {u_{1}}^{- 2} \frac{d}{d x} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.5.3

Replace all occurrences of $u_{1}$ with ${(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}$ .

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} \frac{d}{d x} {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 1296 + {(79 - x)}^{2}$ .

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Step 2.2.6.1

To apply the Chain Rule, set $u_{2}$ as $1296 + {(79 - x)}^{2}$ .

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{d}{d u (2)} ({u_{2}}^{\frac{1}{2}}) \frac{d}{d x} (1296 + {(79 - x)}^{2})))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.6.2

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

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} (1296 + {(79 - x)}^{2})))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.6.3

Replace all occurrences of $u_{2}$ with $1296 + {(79 - x)}^{2}$ .

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} (1296 + {(79 - x)}^{2})))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.7

By the Sum Rule, the derivative of $1296 + {(79 - x)}^{2}$ with respect to $x$ is $\frac{d}{d x} [1296] + \frac{d}{d x} [{(79 - x)}^{2}]$ .

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (\frac{d}{d x} (1296) + \frac{d}{d x} ({(79 - x)}^{2})))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.8

Since $1296$ is constant with respect to $x$ , the derivative of $1296$ with respect to $x$ is $0$ .

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} ({(79 - x)}^{2})))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 79 - x$ .

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Step 2.2.9.1

To apply the Chain Rule, set $u_{3}$ as $79 - x$ .

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + \frac{d}{d u (3)} ({u_{3}}^{2}) \frac{d}{d x} (79 - x)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.9.2

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

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 u_{3} \frac{d}{d x} (79 - x)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.9.3

Replace all occurrences of $u_{3}$ with $79 - x$ .

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) \frac{d}{d x} (79 - x)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.10

By the Sum Rule, the derivative of $79 - x$ with respect to $x$ is $\frac{d}{d x} [79] + \frac{d}{d x} [- x]$ .

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (\frac{d}{d x} (79) + \frac{d}{d x} (- x))))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.11

Since $79$ is constant with respect to $x$ , the derivative of $79$ with respect to $x$ is $0$ .

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (0 + \frac{d}{d x} (- x))))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.12

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

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (0 - \frac{d}{d x} x)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.13

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

$f'' (x) = - ((79 - x) (26 (- {({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (79 - x)) + \frac{d}{d x} (10)$

Step 2.2.14

By the Sum Rule, the derivative of $79 - x$ with respect to $x$ is $\frac{d}{d x} [79] + \frac{d}{d x} [- x]$ .

Step 2.2.15

Since $79$ is constant with respect to $x$ , the derivative of $79$ with respect to $x$ is $0$ .

Step 2.2.16

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

Step 2.2.17

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

Step 2.2.18

Multiply the exponents in ${({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})}^{- 2}$ .

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Step 2.2.18.1

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

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} \cdot - 2} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.18.2

Cancel the common factor of $2$ .

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Step 2.2.18.2.1

Factor $2$ out of $- 2$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.18.2.2

Cancel the common factor.

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.18.2.3

Rewrite the expression.

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.19

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

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.20

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

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.21

Combine the numerators over the common denominator.

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1 - 1 \cdot 2}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.22

Simplify the numerator.

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Step 2.2.22.1

Multiply $- 1$ by $2$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{1 - 2}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.22.2

Subtract $2$ from $1$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{\frac{- 1}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.23

Move the negative in front of the fraction.

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}} (0 + 2 (79 - x) (0 - 1 \cdot 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.24

Multiply $- 1$ by $1$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}} (0 + 2 (79 - x) (0 - 1)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.25

Subtract $1$ from $0$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}} (0 + 2 (79 - x) \cdot - 1))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.26

Multiply $- 1$ by $2$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}} (0 - 2 (79 - x)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.27

Subtract $2 (79 - x)$ from $0$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{2} \cdot ({(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}} (- 2 (79 - x)))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.28

Combine $\frac{1}{2}$ and ${(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}}$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{{(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}}}{2} \cdot (- 2 (79 - x))))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.29

Combine $- 2$ and $\frac{{(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}}}{2}$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{- 2 {(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}}}{2} \cdot (79 - x)))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.30

Move ${(1296 + {(79 - x)}^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{- 2}{2 {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (79 - x)))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.31

Factor $2$ out of $- 2$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{2 \cdot - 1}{2 {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (79 - x)))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.32

Cancel the common factors.

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Step 2.2.32.1

Factor $2$ out of $2 {(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}$ .

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{2 \cdot - 1}{2 ({(1296 + {(79 - x)}^{2})}^{\frac{1}{2}})} \cdot (79 - x)))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.32.2

Cancel the common factor.

Step 2.2.32.3

Rewrite the expression.

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{- 1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (79 - x)))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.33

Move the negative in front of the fraction.

$f'' (x) = - ((79 - x) (26 (- {(1296 + {(79 - x)}^{2})}^{- 1} (- \frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (79 - x)))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.34

Multiply $- 1$ by $- 1$ .

$f'' (x) = - ((79 - x) (26 (1 {(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (79 - x)))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.35

Multiply ${(1296 + {(79 - x)}^{2})}^{- 1}$ by $1$ .

$f'' (x) = - ((79 - x) (26 ({(1296 + {(79 - x)}^{2})}^{- 1} (\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (79 - x)))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.36

Combine $\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}$ and ${(1296 + {(79 - x)}^{2})}^{- 1}$ .

$f'' (x) = - ((79 - x) (26 (\frac{{(1296 + {(79 - x)}^{2})}^{- 1}}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (79 - x))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.37

Move ${(1296 + {(79 - x)}^{2})}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - ((79 - x) (26 (\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}} (1296 + {(79 - x)}^{2})} \cdot (79 - x))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.38

Multiply ${(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}$ by $1296 + {(79 - x)}^{2}$ by adding the exponents.

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Step 2.2.38.1

Multiply ${(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}$ by $1296 + {(79 - x)}^{2}$ .

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Step 2.2.38.1.1

Raise $1296 + {(79 - x)}^{2}$ to the power of $1$ .

Step 2.2.38.1.2

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

$f'' (x) = - ((79 - x) (26 (\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2} + 1}} \cdot (79 - x))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.38.2

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

$f'' (x) = - ((79 - x) (26 (\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2} + \frac{2}{2}}} \cdot (79 - x))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.38.3

Combine the numerators over the common denominator.

$f'' (x) = - ((79 - x) (26 (\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{1 + 2}{2}}} \cdot (79 - x))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.38.4

Add $1$ and $2$ .

$f'' (x) = - ((79 - x) (26 (\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} \cdot (79 - x))) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.39

Combine $\frac{1}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}$ and $26$ .

$f'' (x) = - ((79 - x) (\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} \cdot (79 - x)) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.40

Raise $79 - x$ to the power of $1$ .

$f'' (x) = - ((79 - x) (79 - x) (\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.41

Raise $79 - x$ to the power of $1$ .

$f'' (x) = - ((79 - x) (79 - x) (\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.42

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

$f'' (x) = - ({(79 - x)}^{1 + 1} (\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.43

Add $1$ and $1$ .

$f'' (x) = - ({(79 - x)}^{2} (\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}) + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.44

Combine ${(79 - x)}^{2}$ and $\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}$ .

$f'' (x) = - (\frac{{(79 - x)}^{2} \cdot 26}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.45

Move $26$ to the left of ${(79 - x)}^{2}$ .

$f'' (x) = - (\frac{26 \cdot {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1 \cdot 1)) + \frac{d}{d x} (10)$

Step 2.2.46

Multiply $- 1$ by $1$ .

$f'' (x) = - (\frac{26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot (0 - 1)) + \frac{d}{d x} (10)$

Step 2.2.47

Subtract $1$ from $0$ .

$f'' (x) = - (\frac{26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot - 1) + \frac{d}{d x} (10)$

Step 2.2.48

Combine $\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}$ and $- 1$ .

$f'' (x) = - (\frac{26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + \frac{26 \cdot - 1}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}) + \frac{d}{d x} (10)$

Step 2.2.49

Multiply $26$ by $- 1$ .

$f'' (x) = - (\frac{26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + \frac{- 26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}) + \frac{d}{d x} (10)$

Step 2.2.50

Move the negative in front of the fraction.

$f'' (x) = - (\frac{26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} - \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}) + \frac{d}{d x} (10)$

Step 2.2.51

To write $- \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}{{(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}$ .

$f'' (x) = - (\frac{26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} - \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} \cdot \frac{{(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}{{(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}) + \frac{d}{d x} (10)$

Step 2.2.52

Write each expression with a common denominator of ${(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.52.1

Multiply $\frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}}$ by $\frac{{(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}{{(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}$ .

$f'' (x) = - (\frac{26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} - \frac{26 {(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}} {(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}) + \frac{d}{d x} (10)$

Step 2.2.52.2

Multiply ${(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}$ by ${(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}$ by adding the exponents.

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Step 2.2.52.2.1

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

$f'' (x) = - (\frac{26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} - \frac{26 {(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2} + \frac{2}{2}}}) + \frac{d}{d x} (10)$

Step 2.2.52.2.2

Combine the numerators over the common denominator.

$f'' (x) = - (\frac{26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} - \frac{26 {(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}{{(1296 + {(79 - x)}^{2})}^{\frac{1 + 2}{2}}}) + \frac{d}{d x} (10)$

Step 2.2.52.2.3

Add $1$ and $2$ .

$f'' (x) = - (\frac{26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} - \frac{26 {(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}) + \frac{d}{d x} (10)$

Step 2.2.53

Combine the numerators over the common denominator.

$f'' (x) = - \frac{26 {(79 - x)}^{2} - 26 {(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + \frac{d}{d x} (10)$

Step 2.2.54

Cancel the common factor of $2$ .

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Step 2.2.54.1

Cancel the common factor.

$f'' (x) = - \frac{26 {(79 - x)}^{2} - 26 {(1296 + {(79 - x)}^{2})}^{\frac{2}{2}}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + \frac{d}{d x} (10)$

Step 2.2.54.2

Rewrite the expression.

$f'' (x) = - \frac{26 {(79 - x)}^{2} - 26 (1296 + {(79 - x)}^{2})}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + \frac{d}{d x} (10)$

Step 2.2.55

Simplify.

$f'' (x) = - \frac{26 {(79 - x)}^{2} - 26 (1296 + {(79 - x)}^{2})}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + \frac{d}{d x} (10)$

Step 2.3

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

$f'' (x) = - \frac{26 {(79 - x)}^{2} - 26 (1296 + {(79 - x)}^{2})}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + 0$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = - \frac{26 {(79 - x)}^{2} - 26 \cdot 1296 - 26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + 0$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $- 26$ by $1296$ .

$f'' (x) = - \frac{26 {(79 - x)}^{2} - 33696 - 26 {(79 - x)}^{2}}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + 0$

Step 2.4.2.2

Subtract $26 {(79 - x)}^{2}$ from $26 {(79 - x)}^{2}$ .

$f'' (x) = - \frac{0 - 33696}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + 0$

Step 2.4.2.3

Subtract $33696$ from $0$ .

$f'' (x) = - \frac{- 33696}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + 0$

Step 2.4.2.4

Move the negative in front of the fraction.

$f'' (x) = \frac{33696}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + 0$

Step 2.4.2.5

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{33696}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}) + 0$

Step 2.4.2.6

Multiply $\frac{33696}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}$ by $1$ .

$f'' (x) = \frac{33696}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}} + 0$

Step 2.4.2.7

Add $\frac{33696}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}$ and $0$ .

$f'' (x) = \frac{33696}{{(1296 + {(79 - x)}^{2})}^{\frac{3}{2}}}$

Step 2.4.3

Reorder terms.

$f'' (x) = \frac{33696}{{({(79 - x)}^{2} + 1296)}^{\frac{3}{2}}}$

Step 2.4.4

Simplify the denominator.

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Step 2.4.4.1

Simplify each term.

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Step 2.4.4.1.1

Rewrite ${(79 - x)}^{2}$ as $(79 - x) (79 - x)$ .

$f'' (x) = \frac{33696}{{((79 - x) (79 - x) + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.2

Expand $(79 - x) (79 - x)$ using the FOIL Method.

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Step 2.4.4.1.2.1

Apply the distributive property.

$f'' (x) = \frac{33696}{{(79 (79 - x) - x (79 - x) + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.2.2

Apply the distributive property.

$f'' (x) = \frac{33696}{{(79 \cdot 79 + 79 (- x) - x (79 - x) + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.2.3

Apply the distributive property.

$f'' (x) = \frac{33696}{{(79 \cdot 79 + 79 (- x) - x \cdot 79 - x (- x) + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.3

Simplify and combine like terms.

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Step 2.4.4.1.3.1

Simplify each term.

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Step 2.4.4.1.3.1.1

Multiply $79$ by $79$ .

$f'' (x) = \frac{33696}{{(6241 + 79 (- x) - x \cdot 79 - x (- x) + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.3.1.2

Multiply $- 1$ by $79$ .

$f'' (x) = \frac{33696}{{(6241 - 79 x - x \cdot 79 - x (- x) + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.3.1.3

Multiply $79$ by $- 1$ .

$f'' (x) = \frac{33696}{{(6241 - 79 x - 79 x - x (- x) + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.3.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{33696}{{(6241 - 79 x - 79 x - 1 \cdot (- 1 x \cdot x) + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.3.1.5

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

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Step 2.4.4.1.3.1.5.1

Move $x$ .

$f'' (x) = \frac{33696}{{(6241 - 79 x - 79 x - 1 \cdot (- 1 (x \cdot x)) + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.3.1.5.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{33696}{{(6241 - 79 x - 79 x - 1 \cdot (- 1 x^{2}) + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.3.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = \frac{33696}{{(6241 - 79 x - 79 x + 1 x^{2} + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.3.1.7

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

$f'' (x) = \frac{33696}{{(6241 - 79 x - 79 x + x^{2} + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.1.3.2

Subtract $79 x$ from $- 79 x$ .

$f'' (x) = \frac{33696}{{(6241 - 158 x + x^{2} + 1296)}^{\frac{3}{2}}}$

Step 2.4.4.2

Add $6241$ and $1296$ .

$f'' (x) = \frac{33696}{{(- 158 x + x^{2} + 7537)}^{\frac{3}{2}}}$

Step 3

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

$- (79 - x) \frac{26}{{(1296 + {(79 - x)}^{2})}^{\frac{1}{2}}} + 10 = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6