Calculus Examples

$\frac{136}{1 + 0.25 {(t - 4.5)}^{2}} + 28$

Step 1

Write $\frac{136}{1 + 0.25 {(t - 4.5)}^{2}} + 28$ as a function.

$f (t) = \frac{136}{1 + 0.25 {(t - 4.5)}^{2}} + 28$

Step 2

Find the first derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $\frac{136}{1 + 0.25 {(t - 4.5)}^{2}} + 28$ with respect to $t$ is $\frac{d}{d t} [\frac{136}{1 + 0.25 {(t - 4.5)}^{2}}] + \frac{d}{d t} [28]$ .

$\frac{d}{d t} [\frac{136}{1 + 0.25 {(t - 4.5)}^{2}}] + \frac{d}{d t} [28]$

Step 2.1.2

Evaluate $\frac{d}{d t} [\frac{136}{1 + 0.25 {(t - 4.5)}^{2}}]$ .

Tap for more steps...

Step 2.1.2.1

Since $136$ is constant with respect to $t$ , the derivative of $\frac{136}{1 + 0.25 {(t - 4.5)}^{2}}$ with respect to $t$ is $136 \frac{d}{d t} [\frac{1}{1 + 0.25 {(t - 4.5)}^{2}}]$ .

$136 \frac{d}{d t} [\frac{1}{1 + 0.25 {(t - 4.5)}^{2}}] + \frac{d}{d t} [28]$

Step 2.1.2.2

Rewrite $\frac{1}{1 + 0.25 {(t - 4.5)}^{2}}$ as ${(1 + 0.25 {(t - 4.5)}^{2})}^{- 1}$ .

$136 \frac{d}{d t} [{(1 + 0.25 {(t - 4.5)}^{2})}^{- 1}] + \frac{d}{d t} [28]$

Step 2.1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = 1 + 0.25 {(t - 4.5)}^{2}$ .

Tap for more steps...

Step 2.1.2.3.1

To apply the Chain Rule, set $u_{1}$ as $1 + 0.25 {(t - 4.5)}^{2}$ .

$136 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d t} [1 + 0.25 {(t - 4.5)}^{2}]) + \frac{d}{d t} [28]$

Step 2.1.2.3.2

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

$136 (- {u_{1}}^{- 2} \frac{d}{d t} [1 + 0.25 {(t - 4.5)}^{2}]) + \frac{d}{d t} [28]$

Step 2.1.2.3.3

Replace all occurrences of $u_{1}$ with $1 + 0.25 {(t - 4.5)}^{2}$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} \frac{d}{d t} [1 + 0.25 {(t - 4.5)}^{2}]) + \frac{d}{d t} [28]$

Step 2.1.2.4

By the Sum Rule, the derivative of $1 + 0.25 {(t - 4.5)}^{2}$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [0.25 {(t - 4.5)}^{2}]$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (\frac{d}{d t} [1] + \frac{d}{d t} [0.25 {(t - 4.5)}^{2}])) + \frac{d}{d t} [28]$

Step 2.1.2.5

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

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + \frac{d}{d t} [0.25 {(t - 4.5)}^{2}])) + \frac{d}{d t} [28]$

Step 2.1.2.6

Since $0.25$ is constant with respect to $t$ , the derivative of $0.25 {(t - 4.5)}^{2}$ with respect to $t$ is $0.25 \frac{d}{d t} [{(t - 4.5)}^{2}]$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + 0.25 \frac{d}{d t} [{(t - 4.5)}^{2}])) + \frac{d}{d t} [28]$

Step 2.1.2.7

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = t - 4.5$ .

Tap for more steps...

Step 2.1.2.7.1

To apply the Chain Rule, set $u_{2}$ as $t - 4.5$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + 0.25 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d t} [t - 4.5]))) + \frac{d}{d t} [28]$

Step 2.1.2.7.2

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

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + 0.25 (2 u_{2} \frac{d}{d t} [t - 4.5]))) + \frac{d}{d t} [28]$

Step 2.1.2.7.3

Replace all occurrences of $u_{2}$ with $t - 4.5$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + 0.25 (2 (t - 4.5) \frac{d}{d t} [t - 4.5]))) + \frac{d}{d t} [28]$

Step 2.1.2.8

By the Sum Rule, the derivative of $t - 4.5$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- 4.5]$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + 0.25 (2 (t - 4.5) (\frac{d}{d t} [t] + \frac{d}{d t} [- 4.5])))) + \frac{d}{d t} [28]$

Step 2.1.2.9

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

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + 0.25 (2 (t - 4.5) (1 + \frac{d}{d t} [- 4.5])))) + \frac{d}{d t} [28]$

Step 2.1.2.10

Since $- 4.5$ is constant with respect to $t$ , the derivative of $- 4.5$ with respect to $t$ is $0$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + 0.25 (2 (t - 4.5) (1 + 0)))) + \frac{d}{d t} [28]$

Step 2.1.2.11

Add $1$ and $0$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + 0.25 (2 (t - 4.5) \cdot 1))) + \frac{d}{d t} [28]$

Step 2.1.2.12

Multiply $2$ by $1$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + 0.25 (2 (t - 4.5)))) + \frac{d}{d t} [28]$

Step 2.1.2.13

Multiply $2$ by $0.25$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0 + 0.5 (t - 4.5))) + \frac{d}{d t} [28]$

Step 2.1.2.14

Add $0$ and $0.5 (t - 4.5)$ .

$136 (- {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (0.5 (t - 4.5))) + \frac{d}{d t} [28]$

Step 2.1.2.15

Multiply $0.5$ by $- 1$ .

$136 (- 0.5 {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (t - 4.5)) + \frac{d}{d t} [28]$

Step 2.1.2.16

Multiply $- 0.5$ by $136$ .

$- 68 {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (t - 4.5) + \frac{d}{d t} [28]$

Step 2.1.3

Since $28$ is constant with respect to $t$ , the derivative of $28$ with respect to $t$ is $0$ .

$- 68 {(1 + 0.25 {(t - 4.5)}^{2})}^{- 2} (t - 4.5) + 0$

Step 2.1.4

Simplify.

Tap for more steps...

Step 2.1.4.1

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

$- 68 \frac{1}{{(1 + 0.25 {(t - 4.5)}^{2})}^{2}} (t - 4.5) + 0$

Step 2.1.4.2

Combine terms.

Tap for more steps...

Step 2.1.4.2.1

Combine $- 68$ and $\frac{1}{{(1 + 0.25 {(t - 4.5)}^{2})}^{2}}$ .

$\frac{- 68}{{(1 + 0.25 {(t - 4.5)}^{2})}^{2}} (t - 4.5) + 0$

Step 2.1.4.2.2

Move the negative in front of the fraction.

$- \frac{68}{{(1 + 0.25 {(t - 4.5)}^{2})}^{2}} (t - 4.5) + 0$

Step 2.1.4.2.3

Add $- \frac{68}{{(1 + 0.25 {(t - 4.5)}^{2})}^{2}} (t - 4.5)$ and $0$ .

$- \frac{68}{{(1 + 0.25 {(t - 4.5)}^{2})}^{2}} (t - 4.5)$

Step 2.1.4.3

Reorder the factors of $- \frac{68}{{(1 + 0.25 {(t - 4.5)}^{2})}^{2}} (t - 4.5)$ .

$- (t - 4.5) \frac{68}{{(1 + 0.25 {(t - 4.5)}^{2})}^{2}}$

Step 2.1.4.4

Apply the distributive property.

$(- t - - 4.5) \frac{68}{{(1 + 0.25 {(t - 4.5)}^{2})}^{2}}$

Step 2.1.4.5

Multiply $- 1$ by $- 4.5$ .

$(- t + 4.5) \frac{68}{{(1 + 0.25 {(t - 4.5)}^{2})}^{2}}$

Step 2.1.4.6

Simplify the denominator.

Tap for more steps...

Step 2.1.4.6.1

Rewrite ${(t - 4.5)}^{2}$ as $(t - 4.5) (t - 4.5)$ .

$(- t + 4.5) \frac{68}{{(1 + 0.25 ((t - 4.5) (t - 4.5)))}^{2}}$

Step 2.1.4.6.2

Expand $(t - 4.5) (t - 4.5)$ using the FOIL Method.

Tap for more steps...

Step 2.1.4.6.2.1

Apply the distributive property.

$(- t + 4.5) \frac{68}{{(1 + 0.25 (t (t - 4.5) - 4.5 (t - 4.5)))}^{2}}$

Step 2.1.4.6.2.2

Apply the distributive property.

$(- t + 4.5) \frac{68}{{(1 + 0.25 (t \cdot t + t \cdot - 4.5 - 4.5 (t - 4.5)))}^{2}}$

Step 2.1.4.6.2.3

Apply the distributive property.

$(- t + 4.5) \frac{68}{{(1 + 0.25 (t \cdot t + t \cdot - 4.5 - 4.5 t - 4.5 \cdot - 4.5))}^{2}}$

Step 2.1.4.6.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.4.6.3.1

Simplify each term.

Tap for more steps...

Step 2.1.4.6.3.1.1

Multiply $t$ by $t$ .

$(- t + 4.5) \frac{68}{{(1 + 0.25 (t^{2} + t \cdot - 4.5 - 4.5 t - 4.5 \cdot - 4.5))}^{2}}$

Step 2.1.4.6.3.1.2

Move $- 4.5$ to the left of $t$ .

$(- t + 4.5) \frac{68}{{(1 + 0.25 (t^{2} - 4.5 \cdot t - 4.5 t - 4.5 \cdot - 4.5))}^{2}}$

Step 2.1.4.6.3.1.3

Multiply $- 4.5$ by $- 4.5$ .

$(- t + 4.5) \frac{68}{{(1 + 0.25 (t^{2} - 4.5 t - 4.5 t + 20.25))}^{2}}$

Step 2.1.4.6.3.2

Subtract $4.5 t$ from $- 4.5 t$ .

$(- t + 4.5) \frac{68}{{(1 + 0.25 (t^{2} - 9 t + 20.25))}^{2}}$

Step 2.1.4.6.4

Apply the distributive property.

$(- t + 4.5) \frac{68}{{(1 + 0.25 t^{2} + 0.25 (- 9 t) + 0.25 \cdot 20.25)}^{2}}$

Step 2.1.4.6.5

Simplify.

Tap for more steps...

Step 2.1.4.6.5.1

Multiply $- 9$ by $0.25$ .

$(- t + 4.5) \frac{68}{{(1 + 0.25 t^{2} - 2.25 t + 0.25 \cdot 20.25)}^{2}}$

Step 2.1.4.6.5.2

Multiply $0.25$ by $20.25$ .

$(- t + 4.5) \frac{68}{{(1 + 0.25 t^{2} - 2.25 t + 5.0625)}^{2}}$

Step 2.1.4.6.6

Add $1$ and $5.0625$ .

$(- t + 4.5) \frac{68}{{(0.25 t^{2} - 2.25 t + 6.0625)}^{2}}$

Step 2.1.4.6.7

Factor $0.0625$ out of $0.25 t^{2} - 2.25 t + 6.0625$ .

Tap for more steps...

Step 2.1.4.6.7.1

Factor $0.0625$ out of $0.25 t^{2}$ .

$(- t + 4.5) \frac{68}{{(0.0625 (4 t^{2}) - 2.25 t + 6.0625)}^{2}}$

Step 2.1.4.6.7.2

Factor $0.0625$ out of $- 2.25 t$ .

$(- t + 4.5) \frac{68}{{(0.0625 (4 t^{2}) + 0.0625 (- 36 t) + 6.0625)}^{2}}$

Step 2.1.4.6.7.3

Factor $0.0625$ out of $6.0625$ .

$(- t + 4.5) \frac{68}{{(0.0625 (4 t^{2}) + 0.0625 (- 36 t) + 0.0625 (97))}^{2}}$

Step 2.1.4.6.7.4

Factor $0.0625$ out of $0.0625 (4 t^{2}) + 0.0625 (- 36 t)$ .

$(- t + 4.5) \frac{68}{{(0.0625 (4 t^{2} - 36 t) + 0.0625 (97))}^{2}}$

Step 2.1.4.6.7.5

Factor $0.0625$ out of $0.0625 (4 t^{2} - 36 t) + 0.0625 (97)$ .

$(- t + 4.5) \frac{68}{{(0.0625 (4 t^{2} - 36 t + 97))}^{2}}$

Step 2.1.4.6.8

Apply the product rule to $0.0625 (4 t^{2} - 36 t + 97)$ .

$(- t + 4.5) \frac{68}{{0.0625}^{2} {(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.6.9

Raise $0.0625$ to the power of $2$ .

$(- t + 4.5) \frac{68}{0.00390625 {(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.7

Factor $68$ out of $68$ .

$(- t + 4.5) \frac{68 (1)}{0.00390625 {(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.8

Factor $0.00390625$ out of $0.00390625 {(4 t^{2} - 36 t + 97)}^{2}$ .

$(- t + 4.5) \frac{68 (1)}{0.00390625 ({(4 t^{2} - 36 t + 97)}^{2})}$

Step 2.1.4.9

Separate fractions.

$(- t + 4.5) (\frac{68}{0.00390625} \cdot \frac{1}{{(4 t^{2} - 36 t + 97)}^{2}})$

Step 2.1.4.10

Divide $68$ by $0.00390625$ .

$(- t + 4.5) (17408 \frac{1}{{(4 t^{2} - 36 t + 97)}^{2}})$

Step 2.1.4.11

Combine $17408$ and $\frac{1}{{(4 t^{2} - 36 t + 97)}^{2}}$ .

$(- t + 4.5) \frac{17408}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.12

Multiply $- t + 4.5$ by $\frac{17408}{{(4 t^{2} - 36 t + 97)}^{2}}$ .

$\frac{(- t + 4.5) \cdot 17408}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.13

Simplify the numerator.

Tap for more steps...

Step 2.1.4.13.1

Factor $0.5$ out of $- t + 4.5$ .

Tap for more steps...

Step 2.1.4.13.1.1

Factor $0.5$ out of $- t$ .

$\frac{(0.5 (- 2 t) + 4.5) \cdot 17408}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.13.1.2

Factor $0.5$ out of $4.5$ .

$\frac{(0.5 (- 2 t) + 0.5 (9)) \cdot 17408}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.13.1.3

Factor $0.5$ out of $0.5 (- 2 t) + 0.5 (9)$ .

$\frac{0.5 (- 2 t + 9) \cdot 17408}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.13.2

Multiply $17408$ by $0.5$ .

$\frac{8704 (- 2 t + 9)}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.14

Factor $- 1$ out of $- 2 t$ .

$\frac{8704 (- (2 t) + 9)}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.15

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{8704 (- (2 t) - 1 (- 9))}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.16

Factor $- 1$ out of $- (2 t) - 1 (- 9)$ .

$\frac{8704 (- (2 t - 9))}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.17

Rewrite $- (2 t - 9)$ as $- 1 (2 t - 9)$ .

$\frac{8704 (- 1 (2 t - 9))}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.1.4.18

Move the negative in front of the fraction.

$f' (t) = - \frac{8704 (2 t - 9)}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 2.2

The first derivative of $f (t)$ with respect to $t$ is $- \frac{8704 (2 t - 9)}{{(4 t^{2} - 36 t + 97)}^{2}}$ .

$- \frac{8704 (2 t - 9)}{{(4 t^{2} - 36 t + 97)}^{2}}$

Step 3

Set the first derivative equal to $0$ then solve the equation $- \frac{8704 (2 t - 9)}{{(4 t^{2} - 36 t + 97)}^{2}} = 0$ .

Tap for more steps...

Step 3.1

Set the first derivative equal to $0$ .

$- \frac{8704 (2 t - 9)}{{(4 t^{2} - 36 t + 97)}^{2}} = 0$

Step 3.2

Set the numerator equal to zero.

$8704 (2 t - 9) = 0$

Step 3.3

Solve the equation for $t$ .

Tap for more steps...

Step 3.3.1

Divide each term in $8704 (2 t - 9) = 0$ by $8704$ and simplify.

Tap for more steps...

Step 3.3.1.1

Divide each term in $8704 (2 t - 9) = 0$ by $8704$ .

$\frac{8704 (2 t - 9)}{8704} = \frac{0}{8704}$

Step 3.3.1.2

Simplify the left side.

Tap for more steps...

Step 3.3.1.2.1

Cancel the common factor of $8704$ .

Tap for more steps...

Step 3.3.1.2.1.1

Cancel the common factor.

$\frac{8704 (2 t - 9)}{8704} = \frac{0}{8704}$

Step 3.3.1.2.1.2

Divide $2 t - 9$ by $1$ .

$2 t - 9 = \frac{0}{8704}$

Step 3.3.1.3

Simplify the right side.

Tap for more steps...

Step 3.3.1.3.1

Divide $0$ by $8704$ .

$2 t - 9 = 0$

Step 3.3.2

Add $9$ to both sides of the equation.

$2 t = 9$

Step 3.3.3

Divide each term in $2 t = 9$ by $2$ and simplify.

Tap for more steps...

Step 3.3.3.1

Divide each term in $2 t = 9$ by $2$ .

$\frac{2 t}{2} = \frac{9}{2}$

Step 3.3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{9}{2}$

Step 3.3.3.2.1.2

Divide $t$ by $1$ .

$t = \frac{9}{2}$

Step 4

The values which make the derivative equal to $0$ are $\frac{9}{2}$ .

$\frac{9}{2}$

Step 5

After finding the point that makes the derivative $f' (t) = - \frac{8704 (2 t - 9)}{{(4 t^{2} - 36 t + 97)}^{2}}$ equal to $0$ or undefined, the interval to check where $f (t) = \frac{136}{1 + 0.25 {(t - 4.5)}^{2}} + 28$ is increasing and where it is decreasing is $(- \infty, \frac{9}{2}) \cup (\frac{9}{2}, \infty)$ .

$(- \infty, \frac{9}{2}) \cup (\frac{9}{2}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, \frac{9}{2})$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $t$ with $\frac{7}{2}$ in the expression.

$f' (\frac{7}{2}) = - \frac{8704 (2 (\frac{7}{2}) - 9)}{{(4 {(\frac{7}{2})}^{2} - 36 (\frac{7}{2}) + 97)}^{2}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Multiply $8704$ by $2 (\frac{7}{2}) - 9$ .

$f' (\frac{7}{2}) = - \frac{- 17408}{{(4 {(\frac{7}{2})}^{2} - 36 (\frac{7}{2}) + 97)}^{2}}$

Step 6.2.2

Simplify the denominator.

Tap for more steps...

Step 6.2.2.1

Apply the product rule to $\frac{7}{2}$ .

$f' (\frac{7}{2}) = - \frac{- 17408}{{(4 (\frac{7^{2}}{2^{2}}) - 36 (\frac{7}{2}) + 97)}^{2}}$

Step 6.2.2.2

Raise $7$ to the power of $2$ .

$f' (\frac{7}{2}) = - \frac{- 17408}{{(4 (\frac{49}{2^{2}}) - 36 (\frac{7}{2}) + 97)}^{2}}$

Step 6.2.2.3

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

$f' (\frac{7}{2}) = - \frac{- 17408}{{(4 (\frac{49}{4}) - 36 (\frac{7}{2}) + 97)}^{2}}$

Step 6.2.2.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.2.2.4.1

Cancel the common factor.

$f' (\frac{7}{2}) = - \frac{- 17408}{{(4 (\frac{49}{4}) - 36 (\frac{7}{2}) + 97)}^{2}}$

Step 6.2.2.4.2

Rewrite the expression.

$f' (\frac{7}{2}) = - \frac{- 17408}{{(49 - 36 (\frac{7}{2}) + 97)}^{2}}$

Step 6.2.2.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.2.5.1

Factor $2$ out of $- 36$ .

$f' (\frac{7}{2}) = - \frac{- 17408}{{(49 + 2 (- 18) (\frac{7}{2}) + 97)}^{2}}$

Step 6.2.2.5.2

Cancel the common factor.

$f' (\frac{7}{2}) = - \frac{- 17408}{{(49 + 2 \cdot (- 18 (\frac{7}{2})) + 97)}^{2}}$

Step 6.2.2.5.3

Rewrite the expression.

$f' (\frac{7}{2}) = - \frac{- 17408}{{(49 - 18 \cdot 7 + 97)}^{2}}$

Step 6.2.2.6

Multiply $- 18$ by $7$ .

$f' (\frac{7}{2}) = - \frac{- 17408}{{(49 - 126 + 97)}^{2}}$

Step 6.2.2.7

Subtract $126$ from $49$ .

$f' (\frac{7}{2}) = - \frac{- 17408}{{(- 77 + 97)}^{2}}$

Step 6.2.2.8

Add $- 77$ and $97$ .

$f' (\frac{7}{2}) = - \frac{- 17408}{20^{2}}$

Step 6.2.2.9

Raise $20$ to the power of $2$ .

$f' (\frac{7}{2}) = - \frac{- 17408}{400}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.2.3.1

Cancel the common factor of $- 17408$ and $400$ .

Tap for more steps...

Step 6.2.3.1.1

Factor $16$ out of $- 17408$ .

$f' (\frac{7}{2}) = - \frac{16 (- 1088)}{400}$

Step 6.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 6.2.3.1.2.1

Factor $16$ out of $400$ .

$f' (\frac{7}{2}) = - \frac{16 \cdot - 1088}{16 \cdot 25}$

Step 6.2.3.1.2.2

Cancel the common factor.

$f' (\frac{7}{2}) = - \frac{16 \cdot - 1088}{16 \cdot 25}$

Step 6.2.3.1.2.3

Rewrite the expression.

$f' (\frac{7}{2}) = - \frac{- 1088}{25}$

Step 6.2.3.2

Move the negative in front of the fraction.

$f' (\frac{7}{2}) = \frac{1088}{25}$

Step 6.2.4

The final answer is $\frac{1088}{25}$ .

$\frac{1088}{25}$

Step 6.3

At $t = \frac{7}{2}$ the derivative is $\frac{1088}{25}$ . Since this is positive, the function is increasing on $(- \infty, \frac{9}{2})$ .

Increasing on $(- \infty, \frac{9}{2})$ since $f' (t) > 0$

Step 7

Substitute a value from the interval $(\frac{9}{2}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

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

$f' (\frac{11}{2}) = - \frac{8704 (2 (\frac{11}{2}) - 9)}{{(4 {(\frac{11}{2})}^{2} - 36 (\frac{11}{2}) + 97)}^{2}}$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Multiply $8704$ by $2 (\frac{11}{2}) - 9$ .

$f' (\frac{11}{2}) = - \frac{17408}{{(4 {(\frac{11}{2})}^{2} - 36 (\frac{11}{2}) + 97)}^{2}}$

Step 7.2.2

Simplify the denominator.

Tap for more steps...

Step 7.2.2.1

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

$f' (\frac{11}{2}) = - \frac{17408}{{(4 (\frac{11^{2}}{2^{2}}) - 36 (\frac{11}{2}) + 97)}^{2}}$

Step 7.2.2.2

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

$f' (\frac{11}{2}) = - \frac{17408}{{(4 (\frac{121}{2^{2}}) - 36 (\frac{11}{2}) + 97)}^{2}}$

Step 7.2.2.3

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

$f' (\frac{11}{2}) = - \frac{17408}{{(4 (\frac{121}{4}) - 36 (\frac{11}{2}) + 97)}^{2}}$

Step 7.2.2.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 7.2.2.4.1

Cancel the common factor.

$f' (\frac{11}{2}) = - \frac{17408}{{(4 (\frac{121}{4}) - 36 (\frac{11}{2}) + 97)}^{2}}$

Step 7.2.2.4.2

Rewrite the expression.

$f' (\frac{11}{2}) = - \frac{17408}{{(121 - 36 (\frac{11}{2}) + 97)}^{2}}$

Step 7.2.2.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.2.2.5.1

Factor $2$ out of $- 36$ .

$f' (\frac{11}{2}) = - \frac{17408}{{(121 + 2 (- 18) (\frac{11}{2}) + 97)}^{2}}$

Step 7.2.2.5.2

Cancel the common factor.

$f' (\frac{11}{2}) = - \frac{17408}{{(121 + 2 \cdot (- 18 (\frac{11}{2})) + 97)}^{2}}$

Step 7.2.2.5.3

Rewrite the expression.

$f' (\frac{11}{2}) = - \frac{17408}{{(121 - 18 \cdot 11 + 97)}^{2}}$

Step 7.2.2.6

Multiply $- 18$ by $11$ .

$f' (\frac{11}{2}) = - \frac{17408}{{(121 - 198 + 97)}^{2}}$

Step 7.2.2.7

Subtract $198$ from $121$ .

$f' (\frac{11}{2}) = - \frac{17408}{{(- 77 + 97)}^{2}}$

Step 7.2.2.8

Add $- 77$ and $97$ .

$f' (\frac{11}{2}) = - \frac{17408}{20^{2}}$

Step 7.2.2.9

Raise $20$ to the power of $2$ .

$f' (\frac{11}{2}) = - \frac{17408}{400}$

Step 7.2.3

Cancel the common factor of $17408$ and $400$ .

Tap for more steps...

Step 7.2.3.1

Factor $16$ out of $17408$ .

$f' (\frac{11}{2}) = - \frac{16 (1088)}{400}$

Step 7.2.3.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.2.1

Factor $16$ out of $400$ .

$f' (\frac{11}{2}) = - \frac{16 \cdot 1088}{16 \cdot 25}$

Step 7.2.3.2.2

Cancel the common factor.

$f' (\frac{11}{2}) = - \frac{16 \cdot 1088}{16 \cdot 25}$

Step 7.2.3.2.3

Rewrite the expression.

$f' (\frac{11}{2}) = - \frac{1088}{25}$

Step 7.2.4

The final answer is $- \frac{1088}{25}$ .

$- \frac{1088}{25}$

Step 7.3

At $t = \frac{11}{2}$ the derivative is $- \frac{1088}{25}$ . Since this is negative, the function is decreasing on $(\frac{9}{2}, \infty)$ .

Decreasing on $(\frac{9}{2}, \infty)$ since $f' (t) < 0$

Step 8

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

Increasing on: $(- \infty, \frac{9}{2})$

Decreasing on: $(\frac{9}{2}, \infty)$

Step 9