Calculus Examples

$f (t) = \frac{\sqrt{2 t + 1}}{{(t + 1)}^{3}}$

Step 1

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

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = {(2 t + 1)}^{\frac{1}{2}}$ and $g (t) = {(t + 1)}^{3}$ .

$\frac{{(t + 1)}^{3} \frac{d}{d t} [{(2 t + 1)}^{\frac{1}{2}}] - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{({(t + 1)}^{3})}^{2}}$

Step 3

Multiply the exponents in ${({(t + 1)}^{3})}^{2}$ .

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Step 3.1

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

$\frac{{(t + 1)}^{3} \frac{d}{d t} [{(2 t + 1)}^{\frac{1}{2}}] - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{3 \cdot 2}}$

Step 3.2

Multiply $3$ by $2$ .

$\frac{{(t + 1)}^{3} \frac{d}{d t} [{(2 t + 1)}^{\frac{1}{2}}] - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 4

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

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Step 4.1

To apply the Chain Rule, set $u_{1}$ as $2 t + 1$ .

$\frac{{(t + 1)}^{3} (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 4.2

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

$\frac{{(t + 1)}^{3} (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 4.3

Replace all occurrences of $u_{1}$ with $2 t + 1$ .

$\frac{{(t + 1)}^{3} (\frac{1}{2} {(2 t + 1)}^{\frac{1}{2} - 1} \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 5

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

$\frac{{(t + 1)}^{3} (\frac{1}{2} {(2 t + 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 6

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

$\frac{{(t + 1)}^{3} (\frac{1}{2} {(2 t + 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 7

Combine the numerators over the common denominator.

$\frac{{(t + 1)}^{3} (\frac{1}{2} {(2 t + 1)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$\frac{{(t + 1)}^{3} (\frac{1}{2} {(2 t + 1)}^{\frac{1 - 2}{2}} \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 8.2

Subtract $2$ from $1$ .

$\frac{{(t + 1)}^{3} (\frac{1}{2} {(2 t + 1)}^{\frac{- 1}{2}} \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 9

Combine fractions.

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Step 9.1

Move the negative in front of the fraction.

$\frac{{(t + 1)}^{3} (\frac{1}{2} {(2 t + 1)}^{- \frac{1}{2}} \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 9.2

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

$\frac{{(t + 1)}^{3} (\frac{{(2 t + 1)}^{- \frac{1}{2}}}{2} \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 9.3

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

$\frac{{(t + 1)}^{3} (\frac{1}{2 {(2 t + 1)}^{\frac{1}{2}}} \frac{d}{d t} [2 t + 1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 9.4

Combine $\frac{1}{2 {(2 t + 1)}^{\frac{1}{2}}}$ and ${(t + 1)}^{3}$ .

$\frac{\frac{{(t + 1)}^{3}}{2 {(2 t + 1)}^{\frac{1}{2}}} \frac{d}{d t} [2 t + 1] - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 10

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

$\frac{\frac{{(t + 1)}^{3}}{2 {(2 t + 1)}^{\frac{1}{2}}} (\frac{d}{d t} [2 t] + \frac{d}{d t} [1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 11

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

$\frac{\frac{{(t + 1)}^{3}}{2 {(2 t + 1)}^{\frac{1}{2}}} (2 \frac{d}{d t} [t] + \frac{d}{d t} [1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 12

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

$\frac{\frac{{(t + 1)}^{3}}{2 {(2 t + 1)}^{\frac{1}{2}}} (2 \cdot 1 + \frac{d}{d t} [1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 13

Multiply $2$ by $1$ .

$\frac{\frac{{(t + 1)}^{3}}{2 {(2 t + 1)}^{\frac{1}{2}}} (2 + \frac{d}{d t} [1]) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 14

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

$\frac{\frac{{(t + 1)}^{3}}{2 {(2 t + 1)}^{\frac{1}{2}}} (2 + 0) - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 15

Simplify terms.

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Step 15.1

Add $2$ and $0$ .

$\frac{\frac{{(t + 1)}^{3}}{2 {(2 t + 1)}^{\frac{1}{2}}} \cdot 2 - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 15.2

Combine $\frac{{(t + 1)}^{3}}{2 {(2 t + 1)}^{\frac{1}{2}}}$ and $2$ .

$\frac{\frac{{(t + 1)}^{3} \cdot 2}{2 {(2 t + 1)}^{\frac{1}{2}}} - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 15.3

Move $2$ to the left of ${(t + 1)}^{3}$ .

$\frac{\frac{2 \cdot {(t + 1)}^{3}}{2 {(2 t + 1)}^{\frac{1}{2}}} - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 15.4

Cancel the common factor.

$\frac{\frac{2 {(t + 1)}^{3}}{2 {(2 t + 1)}^{\frac{1}{2}}} - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 15.5

Rewrite the expression.

$\frac{\frac{{(t + 1)}^{3}}{{(2 t + 1)}^{\frac{1}{2}}} - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 16

Multiply $\frac{\frac{{(t + 1)}^{3}}{{(2 t + 1)}^{\frac{1}{2}}} - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$ by $\frac{{(2 t + 1)}^{\frac{1}{2}}}{{(2 t + 1)}^{\frac{1}{2}}}$ .

$\frac{{(2 t + 1)}^{\frac{1}{2}}}{{(2 t + 1)}^{\frac{1}{2}}} \cdot \frac{\frac{{(t + 1)}^{3}}{{(2 t + 1)}^{\frac{1}{2}}} - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}]}{{(t + 1)}^{6}}$

Step 17

Combine.

$\frac{{(2 t + 1)}^{\frac{1}{2}} (\frac{{(t + 1)}^{3}}{{(2 t + 1)}^{\frac{1}{2}}} - {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}])}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 18

Apply the distributive property.

$\frac{{(2 t + 1)}^{\frac{1}{2}} \frac{{(t + 1)}^{3}}{{(2 t + 1)}^{\frac{1}{2}}} + {(2 t + 1)}^{\frac{1}{2}} (- {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}])}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 19

Cancel the common factor of ${(2 t + 1)}^{\frac{1}{2}}$ .

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Step 19.1

Cancel the common factor.

Step 19.2

Rewrite the expression.

$\frac{{(t + 1)}^{3} + {(2 t + 1)}^{\frac{1}{2}} (- {(2 t + 1)}^{\frac{1}{2}} \frac{d}{d t} [{(t + 1)}^{3}])}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 20

Multiply ${(2 t + 1)}^{\frac{1}{2}}$ by ${(2 t + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 20.1

Move ${(2 t + 1)}^{\frac{1}{2}}$ .

$\frac{{(t + 1)}^{3} + {(2 t + 1)}^{\frac{1}{2}} {(2 t + 1)}^{\frac{1}{2}} (- \frac{d}{d t} [{(t + 1)}^{3}])}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 20.2

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

$\frac{{(t + 1)}^{3} + {(2 t + 1)}^{\frac{1}{2} + \frac{1}{2}} (- \frac{d}{d t} [{(t + 1)}^{3}])}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 20.3

Combine the numerators over the common denominator.

$\frac{{(t + 1)}^{3} + {(2 t + 1)}^{\frac{1 + 1}{2}} (- \frac{d}{d t} [{(t + 1)}^{3}])}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 20.4

Add $1$ and $1$ .

$\frac{{(t + 1)}^{3} + {(2 t + 1)}^{\frac{2}{2}} (- \frac{d}{d t} [{(t + 1)}^{3}])}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 20.5

Divide $2$ by $2$ .

$\frac{{(t + 1)}^{3} + {(2 t + 1)}^{1} (- \frac{d}{d t} [{(t + 1)}^{3}])}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 21

Simplify ${(2 t + 1)}^{1} (- \frac{d}{d t} [{(t + 1)}^{3}])$ .

$\frac{{(t + 1)}^{3} + (2 t + 1) (- \frac{d}{d t} [{(t + 1)}^{3}])}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 22

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{3}$ and $g (t) = t + 1$ .

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Step 22.1

To apply the Chain Rule, set $u_{2}$ as $t + 1$ .

$\frac{{(t + 1)}^{3} + (2 t + 1) (- (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d t} [t + 1]))}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 22.2

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

$\frac{{(t + 1)}^{3} + (2 t + 1) (- (3 {u_{2}}^{2} \frac{d}{d t} [t + 1]))}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 22.3

Replace all occurrences of $u_{2}$ with $t + 1$ .

$\frac{{(t + 1)}^{3} + (2 t + 1) (- (3 {(t + 1)}^{2} \frac{d}{d t} [t + 1]))}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 23

Differentiate.

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Step 23.1

Multiply $3$ by $- 1$ .

$\frac{{(t + 1)}^{3} + (2 t + 1) (- 3 ({(t + 1)}^{2} \frac{d}{d t} [t + 1]))}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 23.2

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

$\frac{{(t + 1)}^{3} + (2 t + 1) (- 3 {(t + 1)}^{2} (\frac{d}{d t} [t] + \frac{d}{d t} [1]))}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 23.3

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

$\frac{{(t + 1)}^{3} + (2 t + 1) (- 3 {(t + 1)}^{2} (1 + \frac{d}{d t} [1]))}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 23.4

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

$\frac{{(t + 1)}^{3} + (2 t + 1) (- 3 {(t + 1)}^{2} (1 + 0))}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 23.5

Simplify the expression.

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Step 23.5.1

Add $1$ and $0$ .

$\frac{{(t + 1)}^{3} + (2 t + 1) (- 3 {(t + 1)}^{2} \cdot 1)}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 23.5.2

Multiply $- 3$ by $1$ .

$\frac{{(t + 1)}^{3} + (2 t + 1) (- 3 {(t + 1)}^{2})}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 24

Simplify.

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Step 24.1

Simplify the numerator.

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Step 24.1.1

Factor ${(t + 1)}^{2}$ out of ${(t + 1)}^{3} + (2 t + 1) (- 3 {(t + 1)}^{2})$ .

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Step 24.1.1.1

Factor ${(t + 1)}^{2}$ out of ${(t + 1)}^{3}$ .

$\frac{{(t + 1)}^{2} (t + 1) + (2 t + 1) (- 3 {(t + 1)}^{2})}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 24.1.1.2

Factor ${(t + 1)}^{2}$ out of $(2 t + 1) (- 3 {(t + 1)}^{2})$ .

$\frac{{(t + 1)}^{2} (t + 1) + {(t + 1)}^{2} ((2 t + 1) \cdot (- 3))}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 24.1.1.3

Factor ${(t + 1)}^{2}$ out of ${(t + 1)}^{2} (t + 1) + {(t + 1)}^{2} ((2 t + 1) \cdot (- 3))$ .

$\frac{{(t + 1)}^{2} (t + 1 + (2 t + 1) \cdot (- 3))}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 24.1.2

Apply the distributive property.

$\frac{{(t + 1)}^{2} (t + 1 + 2 t \cdot - 3 + 1 \cdot - 3)}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 24.1.3

Multiply $- 3$ by $2$ .

$\frac{{(t + 1)}^{2} (t + 1 - 6 t + 1 \cdot - 3)}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 24.1.4

Multiply $- 3$ by $1$ .

$\frac{{(t + 1)}^{2} (t + 1 - 6 t - 3)}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 24.1.5

Subtract $6 t$ from $t$ .

$\frac{{(t + 1)}^{2} (- 5 t + 1 - 3)}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 24.1.6

Subtract $3$ from $1$ .

$\frac{{(t + 1)}^{2} (- 5 t - 2)}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}}$

Step 24.2

Cancel the common factors.

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Step 24.2.1

Factor ${(t + 1)}^{2}$ out of ${(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{6}$ .

$\frac{{(t + 1)}^{2} (- 5 t - 2)}{{(t + 1)}^{2} ({(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{4})}$

Step 24.2.2

Cancel the common factor.

$\frac{{(t + 1)}^{2} (- 5 t - 2)}{{(t + 1)}^{2} ({(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{4})}$

Step 24.2.3

Rewrite the expression.

$\frac{- 5 t - 2}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{4}}$

Step 24.3

Factor $- 1$ out of $- 5 t$ .

$\frac{- (5 t) - 2}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{4}}$

Step 24.4

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

$\frac{- (5 t) - 1 (2)}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{4}}$

Step 24.5

Factor $- 1$ out of $- (5 t) - 1 (2)$ .

$\frac{- (5 t + 2)}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{4}}$

Step 24.6

Rewrite $- (5 t + 2)$ as $- 1 (5 t + 2)$ .

$\frac{- 1 (5 t + 2)}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{4}}$

Step 24.7

Move the negative in front of the fraction.

$- \frac{5 t + 2}{{(2 t + 1)}^{\frac{1}{2}} {(t + 1)}^{4}}$