Calculus Examples

$f (x) = 10 - \frac{20}{4 x^{2} - 52 x + 179}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $10 - \frac{20}{4 x^{2} - 52 x + 179}$ with respect to $x$ is $\frac{d}{d x} [10] + \frac{d}{d x} [- \frac{20}{4 x^{2} - 52 x + 179}]$ .

$\frac{d}{d x} [10] + \frac{d}{d x} [- \frac{20}{4 x^{2} - 52 x + 179}]$

Step 1.1.1.2

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

$0 + \frac{d}{d x} [- \frac{20}{4 x^{2} - 52 x + 179}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- \frac{20}{4 x^{2} - 52 x + 179}]$ .

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Step 1.1.2.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- \frac{20}{4 x^{2} - 52 x + 179}$ with respect to $x$ is $- 20 \frac{d}{d x} [\frac{1}{4 x^{2} - 52 x + 179}]$ .

$0 - 20 \frac{d}{d x} [\frac{1}{4 x^{2} - 52 x + 179}]$

Step 1.1.2.2

Rewrite $\frac{1}{4 x^{2} - 52 x + 179}$ as ${(4 x^{2} - 52 x + 179)}^{- 1}$ .

$0 - 20 \frac{d}{d x} [{(4 x^{2} - 52 x + 179)}^{- 1}]$

Step 1.1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 4 x^{2} - 52 x + 179$ .

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Step 1.1.2.3.1

To apply the Chain Rule, set $u$ as $4 x^{2} - 52 x + 179$ .

$0 - 20 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [4 x^{2} - 52 x + 179])$

Step 1.1.2.3.2

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

$0 - 20 (- u^{- 2} \frac{d}{d x} [4 x^{2} - 52 x + 179])$

Step 1.1.2.3.3

Replace all occurrences of $u$ with $4 x^{2} - 52 x + 179$ .

$0 - 20 (- {(4 x^{2} - 52 x + 179)}^{- 2} \frac{d}{d x} [4 x^{2} - 52 x + 179])$

Step 1.1.2.4

By the Sum Rule, the derivative of $4 x^{2} - 52 x + 179$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 52 x] + \frac{d}{d x} [179]$ .

$0 - 20 (- {(4 x^{2} - 52 x + 179)}^{- 2} (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 52 x] + \frac{d}{d x} [179]))$

Step 1.1.2.5

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$0 - 20 (- {(4 x^{2} - 52 x + 179)}^{- 2} (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 52 x] + \frac{d}{d x} [179]))$

Step 1.1.2.6

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

$0 - 20 (- {(4 x^{2} - 52 x + 179)}^{- 2} (4 (2 x) + \frac{d}{d x} [- 52 x] + \frac{d}{d x} [179]))$

Step 1.1.2.7

Since $- 52$ is constant with respect to $x$ , the derivative of $- 52 x$ with respect to $x$ is $- 52 \frac{d}{d x} [x]$ .

$0 - 20 (- {(4 x^{2} - 52 x + 179)}^{- 2} (4 (2 x) - 52 \frac{d}{d x} [x] + \frac{d}{d x} [179]))$

Step 1.1.2.8

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

$0 - 20 (- {(4 x^{2} - 52 x + 179)}^{- 2} (4 (2 x) - 52 \cdot 1 + \frac{d}{d x} [179]))$

Step 1.1.2.9

Since $179$ is constant with respect to $x$ , the derivative of $179$ with respect to $x$ is $0$ .

$0 - 20 (- {(4 x^{2} - 52 x + 179)}^{- 2} (4 (2 x) - 52 \cdot 1 + 0))$

Step 1.1.2.10

Multiply $2$ by $4$ .

$0 - 20 (- {(4 x^{2} - 52 x + 179)}^{- 2} (8 x - 52 \cdot 1 + 0))$

Step 1.1.2.11

Multiply $- 52$ by $1$ .

$0 - 20 (- {(4 x^{2} - 52 x + 179)}^{- 2} (8 x - 52 + 0))$

Step 1.1.2.12

Add $8 x - 52$ and $0$ .

$0 - 20 (- {(4 x^{2} - 52 x + 179)}^{- 2} (8 x - 52))$

Step 1.1.2.13

Multiply $- 1$ by $- 20$ .

$0 + 20 {(4 x^{2} - 52 x + 179)}^{- 2} (8 x - 52)$

Step 1.1.3

Simplify.

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Step 1.1.3.1

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

$0 + 20 \frac{1}{{(4 x^{2} - 52 x + 179)}^{2}} (8 x - 52)$

Step 1.1.3.2

Combine terms.

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Step 1.1.3.2.1

Combine $20$ and $\frac{1}{{(4 x^{2} - 52 x + 179)}^{2}}$ .

$0 + \frac{20}{{(4 x^{2} - 52 x + 179)}^{2}} (8 x - 52)$

Step 1.1.3.2.2

Add $0$ and $\frac{20}{{(4 x^{2} - 52 x + 179)}^{2}} (8 x - 52)$ .

$\frac{20}{{(4 x^{2} - 52 x + 179)}^{2}} (8 x - 52)$

Step 1.1.3.3

Reorder the factors of $\frac{20}{{(4 x^{2} - 52 x + 179)}^{2}} (8 x - 52)$ .

$(8 x - 52) \frac{20}{{(4 x^{2} - 52 x + 179)}^{2}}$

Step 1.1.3.4

Multiply $8 x - 52$ by $\frac{20}{{(4 x^{2} - 52 x + 179)}^{2}}$ .

$\frac{(8 x - 52) \cdot 20}{{(4 x^{2} - 52 x + 179)}^{2}}$

Step 1.1.3.5

Simplify the numerator.

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Step 1.1.3.5.1

Factor $4$ out of $8 x - 52$ .

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Step 1.1.3.5.1.1

Factor $4$ out of $8 x$ .

$\frac{(4 (2 x) - 52) \cdot 20}{{(4 x^{2} - 52 x + 179)}^{2}}$

Step 1.1.3.5.1.2

Factor $4$ out of $- 52$ .

$\frac{(4 (2 x) + 4 (- 13)) \cdot 20}{{(4 x^{2} - 52 x + 179)}^{2}}$

Step 1.1.3.5.1.3

Factor $4$ out of $4 (2 x) + 4 (- 13)$ .

$\frac{4 (2 x - 13) \cdot 20}{{(4 x^{2} - 52 x + 179)}^{2}}$

Step 1.1.3.5.2

Multiply $20$ by $4$ .

$f' (x) = \frac{80 (2 x - 13)}{{(4 x^{2} - 52 x + 179)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{80 (2 x - 13)}{{(4 x^{2} - 52 x + 179)}^{2}}$ .

$\frac{80 (2 x - 13)}{{(4 x^{2} - 52 x + 179)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{80 (2 x - 13)}{{(4 x^{2} - 52 x + 179)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{80 (2 x - 13)}{{(4 x^{2} - 52 x + 179)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$80 (2 x - 13) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $80 (2 x - 13) = 0$ by $80$ and simplify.

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Step 2.3.1.1

Divide each term in $80 (2 x - 13) = 0$ by $80$ .

$\frac{80 (2 x - 13)}{80} = \frac{0}{80}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $80$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{80 (2 x - 13)}{80} = \frac{0}{80}$

Step 2.3.1.2.1.2

Divide $2 x - 13$ by $1$ .

$2 x - 13 = \frac{0}{80}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $80$ .

$2 x - 13 = 0$

Step 2.3.2

Add $13$ to both sides of the equation.

$2 x = 13$

Step 2.3.3

Divide each term in $2 x = 13$ by $2$ and simplify.

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Step 2.3.3.1

Divide each term in $2 x = 13$ by $2$ .

$\frac{2 x}{2} = \frac{13}{2}$

Step 2.3.3.2

Simplify the left side.

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Step 2.3.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{13}{2}$

Step 2.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{2}$

Step 3

The values which make the derivative equal to $0$ are $\frac{13}{2}$ .

$\frac{13}{2}$

Step 4

After finding the point that makes the derivative $f' (x) = \frac{80 (2 x - 13)}{{(4 x^{2} - 52 x + 179)}^{2}}$ equal to $0$ or undefined, the interval to check where $f (x) = 10 - \frac{20}{4 x^{2} - 52 x + 179}$ is increasing and where it is decreasing is $(- \infty, \frac{13}{2}) \cup (\frac{13}{2}, \infty)$ .

$(- \infty, \frac{13}{2}) \cup (\frac{13}{2}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{13}{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $\frac{11}{2}$ in the expression.

$f' (\frac{11}{2}) = \frac{80 (2 (\frac{11}{2}) - 13)}{{(4 {(\frac{11}{2})}^{2} - 52 (\frac{11}{2}) + 179)}^{2}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Cancel the common factor.

$f' (\frac{11}{2}) = \frac{80 (2 (\frac{11}{2}) - 13)}{{(4 {(\frac{11}{2})}^{2} - 52 (\frac{11}{2}) + 179)}^{2}}$

Step 5.2.1.1.2

Rewrite the expression.

$f' (\frac{11}{2}) = \frac{80 (11 - 13)}{{(4 {(\frac{11}{2})}^{2} - 52 (\frac{11}{2}) + 179)}^{2}}$

Step 5.2.1.2

Subtract $13$ from $11$ .

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(4 {(\frac{11}{2})}^{2} - 52 (\frac{11}{2}) + 179)}^{2}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Apply the product rule to $\frac{11}{2}$ .

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(4 (\frac{11^{2}}{2^{2}}) - 52 (\frac{11}{2}) + 179)}^{2}}$

Step 5.2.2.2

Raise $11$ to the power of $2$ .

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(4 (\frac{121}{2^{2}}) - 52 (\frac{11}{2}) + 179)}^{2}}$

Step 5.2.2.3

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

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(4 (\frac{121}{4}) - 52 (\frac{11}{2}) + 179)}^{2}}$

Step 5.2.2.4

Cancel the common factor of $4$ .

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Step 5.2.2.4.1

Cancel the common factor.

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(4 (\frac{121}{4}) - 52 (\frac{11}{2}) + 179)}^{2}}$

Step 5.2.2.4.2

Rewrite the expression.

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(121 - 52 (\frac{11}{2}) + 179)}^{2}}$

Step 5.2.2.5

Cancel the common factor of $2$ .

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Step 5.2.2.5.1

Factor $2$ out of $- 52$ .

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(121 + 2 (- 26) (\frac{11}{2}) + 179)}^{2}}$

Step 5.2.2.5.2

Cancel the common factor.

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(121 + 2 \cdot (- 26 (\frac{11}{2})) + 179)}^{2}}$

Step 5.2.2.5.3

Rewrite the expression.

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(121 - 26 \cdot 11 + 179)}^{2}}$

Step 5.2.2.6

Multiply $- 26$ by $11$ .

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(121 - 286 + 179)}^{2}}$

Step 5.2.2.7

Subtract $286$ from $121$ .

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{{(- 165 + 179)}^{2}}$

Step 5.2.2.8

Add $- 165$ and $179$ .

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{14^{2}}$

Step 5.2.2.9

Raise $14$ to the power of $2$ .

$f' (\frac{11}{2}) = \frac{80 \cdot - 2}{196}$

Step 5.2.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.1

Multiply $80$ by $- 2$ .

$f' (\frac{11}{2}) = \frac{- 160}{196}$

Step 5.2.3.2

Cancel the common factor of $- 160$ and $196$ .

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Step 5.2.3.2.1

Factor $4$ out of $- 160$ .

$f' (\frac{11}{2}) = \frac{4 (- 40)}{196}$

Step 5.2.3.2.2

Cancel the common factors.

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Step 5.2.3.2.2.1

Factor $4$ out of $196$ .

$f' (\frac{11}{2}) = \frac{4 \cdot - 40}{4 \cdot 49}$

Step 5.2.3.2.2.2

Cancel the common factor.

$f' (\frac{11}{2}) = \frac{4 \cdot - 40}{4 \cdot 49}$

Step 5.2.3.2.2.3

Rewrite the expression.

$f' (\frac{11}{2}) = \frac{- 40}{49}$

Step 5.2.3.3

Move the negative in front of the fraction.

$f' (\frac{11}{2}) = - \frac{40}{49}$

Step 5.2.4

The final answer is $- \frac{40}{49}$ .

$- \frac{40}{49}$

Step 5.3

At $x = \frac{11}{2}$ the derivative is $- \frac{40}{49}$ . Since this is negative, the function is decreasing on $(- \infty, \frac{13}{2})$ .

Decreasing on $(- \infty, \frac{13}{2})$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(\frac{13}{2}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $\frac{15}{2}$ in the expression.

$f' (\frac{15}{2}) = \frac{80 (2 (\frac{15}{2}) - 13)}{{(4 {(\frac{15}{2})}^{2} - 52 (\frac{15}{2}) + 179)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1

Cancel the common factor.

$f' (\frac{15}{2}) = \frac{80 (2 (\frac{15}{2}) - 13)}{{(4 {(\frac{15}{2})}^{2} - 52 (\frac{15}{2}) + 179)}^{2}}$

Step 6.2.1.1.2

Rewrite the expression.

$f' (\frac{15}{2}) = \frac{80 (15 - 13)}{{(4 {(\frac{15}{2})}^{2} - 52 (\frac{15}{2}) + 179)}^{2}}$

Step 6.2.1.2

Subtract $13$ from $15$ .

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(4 {(\frac{15}{2})}^{2} - 52 (\frac{15}{2}) + 179)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Apply the product rule to $\frac{15}{2}$ .

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(4 (\frac{15^{2}}{2^{2}}) - 52 (\frac{15}{2}) + 179)}^{2}}$

Step 6.2.2.2

Raise $15$ to the power of $2$ .

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(4 (\frac{225}{2^{2}}) - 52 (\frac{15}{2}) + 179)}^{2}}$

Step 6.2.2.3

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

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(4 (\frac{225}{4}) - 52 (\frac{15}{2}) + 179)}^{2}}$

Step 6.2.2.4

Cancel the common factor of $4$ .

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Step 6.2.2.4.1

Cancel the common factor.

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(4 (\frac{225}{4}) - 52 (\frac{15}{2}) + 179)}^{2}}$

Step 6.2.2.4.2

Rewrite the expression.

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(225 - 52 (\frac{15}{2}) + 179)}^{2}}$

Step 6.2.2.5

Cancel the common factor of $2$ .

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Step 6.2.2.5.1

Factor $2$ out of $- 52$ .

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(225 + 2 (- 26) (\frac{15}{2}) + 179)}^{2}}$

Step 6.2.2.5.2

Cancel the common factor.

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(225 + 2 \cdot (- 26 (\frac{15}{2})) + 179)}^{2}}$

Step 6.2.2.5.3

Rewrite the expression.

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(225 - 26 \cdot 15 + 179)}^{2}}$

Step 6.2.2.6

Multiply $- 26$ by $15$ .

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(225 - 390 + 179)}^{2}}$

Step 6.2.2.7

Subtract $390$ from $225$ .

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{{(- 165 + 179)}^{2}}$

Step 6.2.2.8

Add $- 165$ and $179$ .

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{14^{2}}$

Step 6.2.2.9

Raise $14$ to the power of $2$ .

$f' (\frac{15}{2}) = \frac{80 \cdot 2}{196}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Multiply $80$ by $2$ .

$f' (\frac{15}{2}) = \frac{160}{196}$

Step 6.2.3.2

Cancel the common factor of $160$ and $196$ .

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Step 6.2.3.2.1

Factor $4$ out of $160$ .

$f' (\frac{15}{2}) = \frac{4 (40)}{196}$

Step 6.2.3.2.2

Cancel the common factors.

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Step 6.2.3.2.2.1

Factor $4$ out of $196$ .

$f' (\frac{15}{2}) = \frac{4 \cdot 40}{4 \cdot 49}$

Step 6.2.3.2.2.2

Cancel the common factor.

$f' (\frac{15}{2}) = \frac{4 \cdot 40}{4 \cdot 49}$

Step 6.2.3.2.2.3

Rewrite the expression.

$f' (\frac{15}{2}) = \frac{40}{49}$

Step 6.2.4

The final answer is $\frac{40}{49}$ .

$\frac{40}{49}$

Step 6.3

At $x = \frac{15}{2}$ the derivative is $\frac{40}{49}$ . Since this is positive, the function is increasing on $(\frac{13}{2}, \infty)$ .

Increasing on $(\frac{13}{2}, \infty)$ since $f' (x) > 0$

Step 7

List the intervals on which the function is increasing and decreasing.

Increasing on: $(\frac{13}{2}, \infty)$

Decreasing on: $(- \infty, \frac{13}{2})$

Step 8